Affine transformation From Wikipedia, the free encyclopedia Contents

1 2 × 2 real matrices 1 1.1 Profile ...... 1 1.2 Equi-areal mapping ...... 2 1.3 Functions of 2 × 2 real matrices ...... 2 1.4 2 × 2 real matrices as complex numbers ...... 3 1.5 References ...... 4

2 3D projection 5 2.1 Orthographic projection ...... 5 2.2 Weak perspective projection ...... 5 2.3 Perspective projection ...... 6 2.4 Diagram ...... 8 2.5 See also ...... 8 2.6 References ...... 9 2.7 External links ...... 9 2.8 Further reading ...... 9

3 Affine coordinate system 10 3.1 See also ...... 10

4 Affine geometry 11 4.1 History ...... 12 4.2 Systems of axioms ...... 12 4.2.1 Pappus’ law ...... 12 4.2.2 Ordered structure ...... 13 4.2.3 Ternary rings ...... 13 4.3 Affine transformations ...... 14 4.4 Affine space ...... 14 4.5 Projective view ...... 15 4.6 See also ...... 15 4.7 References ...... 15 4.8 Further reading ...... 16 4.9 External links ...... 16

i ii CONTENTS

5 Affine group 17 5.1 Relation to general linear group ...... 17 5.1.1 Construction from general linear group ...... 17 5.1.2 Stabilizer of a point ...... 17 5.2 Matrix representation ...... 18 5.3 Planar affine group ...... 18 5.4 Other affine groups ...... 18 5.4.1 General case ...... 18 5.4.2 Special affine group ...... 19 5.4.3 Projective subgroup ...... 19 5.4.4 Poincaré group ...... 19 5.5 See also ...... 19 5.6 References ...... 19

6 Affine space 20 6.1 Informal descriptions ...... 21 6.2 Definition ...... 21 6.2.1 Subtraction and Weyl’s axioms ...... 22 6.2.2 Affine combinations ...... 22 6.3 Examples ...... 22 6.4 Affine subspaces ...... 23 6.5 Affine combinations and affine dependence ...... 23 6.6 Geometric objects as points and vectors ...... 24 6.7 Axioms ...... 24 6.8 Relation to projective spaces ...... 24 6.9 See also ...... 25 6.10 Notes ...... 25 6.11 References ...... 25

7 Affine transformation 27 7.1 Mathematical definition ...... 27 7.1.1 Alternative definition ...... 29 7.2 Representation ...... 29 7.2.1 Augmented matrix ...... 29 7.3 Properties ...... 31 7.4 Affine transformation of the plane ...... 31 7.5 Examples of affine transformations ...... 32 7.5.1 Affine transformations over the real numbers ...... 32 7.5.2 Affine transformation over a finite field ...... 32 7.5.3 Affine transformation in plane geometry ...... 33 7.6 See also ...... 33 7.7 Notes ...... 33 CONTENTS iii

7.8 References ...... 34 7.9 External links ...... 35

8 Augmented matrix 37 8.1 Examples ...... 37 8.1.1 Matrix inverse ...... 37 8.1.2 Existence and number of solutions ...... 38 8.1.3 Solution of a linear system ...... 38 8.2 References ...... 39

9 Barycenter 40 9.1 Two-body problem ...... 41 9.1.1 Primary–secondary examples ...... 41 9.1.2 Inside or outside the Sun? ...... 41 9.2 Gallery ...... 43 9.3 Relativistic corrections ...... 43 9.4 Selected barycentric orbital elements ...... 43 9.5 See also ...... 43 9.6 References ...... 44

10 Bent function 45 10.1 Walsh transform ...... 45 10.2 Definition and properties ...... 46 10.3 Applications ...... 47 10.4 Generalizations ...... 47 10.5 References ...... 48 10.6 Further reading ...... 48

11 Cartesian coordinate system 50 11.1 History ...... 52 11.2 Description ...... 52 11.2.1 One dimension ...... 52 11.2.2 Two dimensions ...... 52 11.2.3 Three dimensions ...... 53 11.2.4 Higher dimensions ...... 54 11.2.5 Generalizations ...... 54 11.3 Notations and conventions ...... 55 11.3.1 Quadrants and octants ...... 55 11.4 Cartesian formulae for the plane ...... 55 11.4.1 Distance between two points ...... 55 11.4.2 Euclidean transformations ...... 56 11.5 Orientation and handedness ...... 59 11.5.1 In two dimensions ...... 59 iv CONTENTS

11.5.2 In three dimensions ...... 60 11.6 Representing a vector in the standard basis ...... 62 11.7 Applications ...... 62 11.8 See also ...... 63 11.9 Notes ...... 63 11.10References ...... 63 11.11Sources ...... 63 11.12Further reading ...... 64 11.13External links ...... 64

12 Collinearity 65 12.1 Points on a line ...... 65 12.2 Examples in Euclidean geometry ...... 65 12.2.1 Triangles ...... 65 12.2.2 Quadrilaterals ...... 66 12.2.3 Hexagons ...... 66 12.2.4 Conic sections ...... 66 12.2.5 Cones ...... 67 12.2.6 Tetrahedrons ...... 67 12.3 Algebra ...... 67 12.3.1 Collinearity of points whose coordinates are given ...... 67 12.3.2 Collinearity of points whose pairwise distances are given ...... 67 12.4 Number theory ...... 68 12.5 Concurrency (plane dual) ...... 68 12.6 Collinearity graph ...... 68 12.7 Usage in statistics and econometrics ...... 68 12.8 Usage in other areas ...... 68 12.8.1 Antenna arrays ...... 69 12.8.2 Photography ...... 70 12.9 See also ...... 70 12.10Notes ...... 70 12.11References ...... 70

13 Endianness 71 13.1 Illustration ...... 71 13.2 History ...... 71 13.2.1 Etymology ...... 72 13.3 Hardware ...... 72 13.3.1 Bi-endian hardware ...... 74 13.3.2 Floating-point ...... 74 13.4 Optimization ...... 74 13.4.1 Calculation order ...... 74 CONTENTS v

13.5 Mapping multi-byte binary values to memory ...... 75 13.6 Examples ...... 75 13.6.1 Big-endian ...... 75 13.6.2 Little-endian ...... 75 13.6.3 Middle-endian ...... 77 13.7 Networking ...... 78 13.8 Files and byte swap ...... 78 13.9 Bit endianness ...... 79 13.10References ...... 79 13.11Further reading ...... 80 13.12External links ...... 80

14 Euclidean distance 81 14.1 Definition ...... 81 14.1.1 One dimension ...... 82 14.1.2 Two dimensions ...... 82 14.1.3 Three dimensions ...... 82 14.1.4 n dimensions ...... 82 14.1.5 Squared Euclidean distance ...... 82 14.2 See also ...... 83 14.3 References ...... 83

15 Euclidean space 84 15.1 Intuitive overview ...... 84 15.2 Euclidean structure ...... 86 15.2.1 Distance ...... 86 15.2.2 Angle ...... 87 15.2.3 Rotations and reflections ...... 87 15.2.4 Euclidean group ...... 89 15.3 Non-Cartesian coordinates ...... 89 15.4 Geometric shapes ...... 89 15.4.1 Lines, planes, and other subspaces ...... 90 15.4.2 Line segments and triangles ...... 91 15.4.3 Polytopes and root systems ...... 92 15.4.4 Curves ...... 92 15.4.5 Balls, spheres, and hypersurfaces ...... 92 15.5 Topology ...... 93 15.6 Applications ...... 93 15.7 Alternatives and generalizations ...... 93 15.7.1 Curved spaces ...... 93 15.7.2 Indefinite quadratic form ...... 93 15.7.3 Other number fields ...... 94 vi CONTENTS

15.7.4 Infinite dimensions ...... 94 15.8 See also ...... 94 15.9 Footnotes ...... 94 15.10References ...... 94 15.11External links ...... 94

16 Flat (geometry) 95 16.1 Descriptions ...... 95 16.1.1 By equations ...... 95 16.1.2 Parametric ...... 95 16.2 Operations and relations on flats ...... 96 16.2.1 Intersecting, parallel, and skew flats ...... 96 16.2.2 Join ...... 96 16.2.3 Properties of operations ...... 96 16.3 Euclidean geometry ...... 96 16.4 See also ...... 96 16.5 Notes ...... 97 16.6 References ...... 97 16.7 External links ...... 97

17 Function composition 98 17.1 Examples ...... 98 17.2 Properties ...... 99 17.3 Composition monoids ...... 101 17.4 Functional powers ...... 101 17.5 Alternative notations ...... 102 17.6 Composition operator ...... 103 17.7 In programming languages ...... 103 17.8 Multivariate functions ...... 103 17.9 Generalizations ...... 104 17.10Typography ...... 104 17.11See also ...... 104 17.12Notes ...... 105 17.13References ...... 105 17.14External links ...... 105

18 General linear group 106 18.1 General linear group of a vector space ...... 106 18.2 In terms of determinants ...... 107 18.3 As a Lie group ...... 107 18.3.1 Real case ...... 107 18.3.2 Complex case ...... 107 CONTENTS vii

18.4 Over finite fields ...... 108 18.4.1 History ...... 109 18.5 Special linear group ...... 109 18.6 Other subgroups ...... 109 18.6.1 Diagonal subgroups ...... 109 18.6.2 Classical groups ...... 110 18.7 Related groups and monoids ...... 110 18.7.1 Projective linear group ...... 110 18.7.2 Affine group ...... 110 18.7.3 General semilinear group ...... 110 18.7.4 Full linear monoid ...... 111 18.8 Infinite general linear group ...... 111 18.9 See also ...... 111 18.10Notes ...... 111 18.11External links ...... 112

19 Glide reflection 113 19.1 Description ...... 113 19.2 Wallpaper groups ...... 114 19.3 Glide reflection in nature and games ...... 114 19.4 See also ...... 115 19.5 References ...... 115 19.6 External links ...... 115

20 Homography 116 20.1 Geometric motivation ...... 116 20.2 Definition and expression in homogeneous coordinates ...... 117 20.3 Homographies of a projective line ...... 118 20.4 Projective frame and coordinates ...... 118 20.5 Central collineations ...... 120 20.6 Fundamental theorem of projective geometry ...... 121 20.7 Homography groups ...... 121 20.8 Cross-ratio ...... 122 20.9 Over a ring ...... 122 20.10Periodic homographies ...... 122 20.11See also ...... 122 20.12Notes ...... 123 20.13References ...... 123 20.14Further reading ...... 123

21 Homothetic transformation 124 21.1 Homothety and uniform scaling ...... 124 viii CONTENTS

21.2 See also ...... 125 21.3 Notes ...... 125 21.4 References ...... 125 21.5 External links ...... 125

22 Improper rotation 126 22.1 See also ...... 127 22.2 References ...... 127

23 Inverse function 128 23.1 Definitions ...... 128 23.1.1 Example: squaring and square root functions ...... 128 23.1.2 Inverses in higher mathematics ...... 128 23.1.3 Inverses and composition ...... 129 23.1.4 Note on notation ...... 129 23.2 Properties ...... 129 23.2.1 Uniqueness ...... 129 23.2.2 Symmetry ...... 129 23.2.3 Self-inverses ...... 130 23.3 Inverses in calculus ...... 130 23.3.1 Formula for the inverse ...... 130 23.3.2 Graph of the inverse ...... 131 23.3.3 Inverses and derivatives ...... 131 23.4 Real-world examples ...... 132 23.5 Generalizations ...... 132 23.5.1 Partial inverses ...... 133 23.5.2 Left and right inverses ...... 133 23.5.3 Preimages ...... 134 23.6 See also ...... 134 23.7 Notes ...... 134 23.8 References ...... 135 23.9 Further reading ...... 135 23.10External links ...... 135

24 Linear function (calculus) 142 24.1 Properties ...... 142 24.2 Slope ...... 142 24.3 Relationship with linear equations ...... 143 24.4 Relationship with other classes of functions ...... 146 24.5 Notes ...... 147 24.6 See also ...... 147 24.7 References ...... 147 CONTENTS ix

24.8 External links ...... 147

25 Linear map 148 25.1 Definition and first consequences ...... 148 25.2 Examples ...... 149 25.3 Matrices ...... 149 25.4 Examples of linear transformation matrices ...... 151 25.5 Forming new linear maps from given ones ...... 152 25.6 Endomorphisms and automorphisms ...... 152 25.7 Kernel, image and the rank–nullity theorem ...... 152 25.8 Cokernel ...... 153 25.8.1 Index ...... 154 25.9 Algebraic classifications of linear transformations ...... 154 25.10Change of basis ...... 155 25.11Continuity ...... 155 25.12Applications ...... 155 25.13See also ...... 155 25.14Notes ...... 156 25.15References ...... 156

26 Matrix (mathematics) 157 26.1 Definition ...... 158 26.1.1 Size ...... 158 26.2 Notation ...... 159 26.3 Basic operations ...... 159 26.3.1 Addition, scalar multiplication and transposition ...... 159 26.3.2 Matrix multiplication ...... 160 26.3.3 Row operations ...... 161 26.3.4 Submatrix ...... 161 26.4 Linear equations ...... 161 26.5 Linear transformations ...... 162 26.6 Square matrices ...... 162 26.6.1 Main types ...... 162 26.6.2 Main operations ...... 165 26.7 Computational aspects ...... 166 26.8 Decomposition ...... 167 26.9 Abstract algebraic aspects and generalizations ...... 168 26.9.1 Matrices with more general entries ...... 168 26.9.2 Relationship to linear maps ...... 169 26.9.3 Matrix groups ...... 169 26.9.4 Infinite matrices ...... 170 26.9.5 Empty matrices ...... 170 x CONTENTS

26.10Applications ...... 171 26.10.1 Graph theory ...... 171 26.10.2 Analysis and geometry ...... 171 26.10.3 Probability theory and statistics ...... 172 26.10.4 Symmetries and transformations in physics ...... 173 26.10.5 Linear combinations of quantum states ...... 174 26.10.6 Normal modes ...... 174 26.10.7 Geometrical optics ...... 175 26.10.8 Electronics ...... 175 26.11History ...... 175 26.11.1 Other historical usages of the word “matrix” in mathematics ...... 176 26.12See also ...... 176 26.13Notes ...... 177 26.14References ...... 180 26.14.1 Physics references ...... 182 26.14.2 Historical references ...... 183 26.15External links ...... 183

27 Origin (mathematics) 185 27.1 Cartesian coordinates ...... 186 27.2 Other coordinate systems ...... 186 27.3 See also ...... 186 27.4 References ...... 186

28 Orthogonal matrix 187 28.1 Overview ...... 187 28.2 Examples ...... 188 28.3 Elementary constructions ...... 188 28.3.1 Lower dimensions ...... 188 28.3.2 Higher dimensions ...... 189 28.3.3 Primitives ...... 189 28.4 Properties ...... 190 28.4.1 Matrix properties ...... 190 28.4.2 Group properties ...... 190 28.4.3 Canonical form ...... 191 28.4.4 Lie algebra ...... 191 28.5 Numerical linear algebra ...... 192 28.5.1 Benefits ...... 192 28.5.2 Decompositions ...... 192 28.5.3 Randomization ...... 193 28.5.4 Nearest orthogonal matrix ...... 194 28.6 Spin and pin ...... 194 CONTENTS xi

28.7 Rectangular matrices ...... 194 28.8 See also ...... 194 28.9 Notes ...... 195 28.10References ...... 195 28.11External links ...... 195

29 Outline of calculus 196 29.1 Branches of calculus ...... 196 29.2 History of calculus ...... 196 29.3 General calculus concepts ...... 196 29.4 Calculus scholars ...... 197 29.5 Calculus lists ...... 197 29.6 See also ...... 197 29.7 References ...... 197 29.8 External links ...... 197

30 Parallelogram 198 30.1 Special cases ...... 198 30.2 Characterizations ...... 198 30.3 Properties ...... 199 30.4 Area formula ...... 200 30.4.1 Another parallelogram with the same base and height ...... 201 30.4.2 Area in terms of Cartesian coordinates of vertices ...... 202 30.5 Proof that diagonals bisect each other ...... 202 30.6 Parallelograms arising from other figures ...... 203 30.6.1 Automedian triangle ...... 203 30.6.2 Varignon parallelogram ...... 203 30.6.3 Tangent parallelogram of an ellipse ...... 203 30.6.4 Faces of a parallelepiped ...... 204 30.7 See also ...... 204 30.8 References ...... 204 30.9 External links ...... 204

31 Position (vector) 205 31.1 Definition ...... 205 31.1.1 Three dimensions ...... 205 31.1.2 n dimensions ...... 205 31.2 Applications ...... 207 31.2.1 Differential geometry ...... 207 31.2.2 Mechanics ...... 207 31.3 Derivatives of position ...... 207 31.4 Relationship to displacement vectors ...... 208 xii CONTENTS

31.5 See also ...... 208 31.6 Notes ...... 208 31.7 References ...... 208

32 Reflection (mathematics) 209 32.1 Construction ...... 209 32.2 Properties ...... 209 32.3 Reflection across a line in the plane ...... 210 32.4 Reflection through a hyperplane in n dimensions ...... 211 32.5 See also ...... 213 32.6 Notes ...... 213 32.7 References ...... 213 32.8 External links ...... 213

33 Rigid body 214 33.1 Kinematics ...... 215 33.1.1 Linear and angular position ...... 215 33.1.2 Linear and angular velocity ...... 215 33.2 Kinematical equations ...... 215 33.2.1 Addition theorem for angular velocity ...... 216 33.2.2 Addition theorem for position ...... 216 33.2.3 Mathematical definition of velocity ...... 216 33.2.4 Mathematical definition of acceleration ...... 216 33.2.5 Velocity of two points fixed on a rigid body ...... 216 33.2.6 Acceleration of two points fixed on a rigid body ...... 216 33.2.7 Angular velocity and acceleration of two points fixed on a rigid body ...... 217 33.2.8 Velocity of one point moving on a rigid body ...... 217 33.2.9 Acceleration of one point moving on a rigid body ...... 217 33.2.10 Other quantities ...... 217 33.3 Kinetics ...... 218 33.4 Geometry ...... 218 33.5 Configuration space ...... 219 33.6 See also ...... 219 33.7 Notes ...... 219 33.8 References ...... 220

34 Rotation 221 34.1 Mathematics ...... 221 34.2 Astronomy ...... 221 34.2.1 Rotation and revolution ...... 223 34.2.2 Retrograde rotation ...... 223 34.3 Physics ...... 224 CONTENTS xiii

34.3.1 Cosmological principle ...... 226 34.3.2 Euler rotations ...... 226 34.4 Flight dynamics ...... 226 34.5 Amusement rides ...... 226 34.6 Sports ...... 226 34.7 See also ...... 228 34.8 References ...... 228 34.9 External links ...... 229

35 Scaling (geometry) 230 35.1 Matrix representation ...... 230 35.1.1 Scaling in arbitrary dimensions ...... 231 35.2 Using homogeneous coordinates ...... 231 35.3 Footnotes ...... 232 35.4 See also ...... 232 35.5 External links ...... 232

36 Semidirect product 233 36.1 Some equivalent definitions of inner semidirect products ...... 233 36.2 Outer semidirect products ...... 233 36.3 Examples ...... 234 36.4 Properties ...... 235 36.4.1 Relation to direct products ...... 235 36.4.2 Non-uniqueness of semidirect products (and further examples) ...... 236 36.4.3 Existence ...... 236 36.4.4 When are they distinct? ...... 236 36.5 Generalizations ...... 236 36.5.1 Groupoids ...... 237 36.5.2 Abelian categories ...... 237 36.6 Notation ...... 237 36.7 See also ...... 237 36.8 Notes ...... 237 36.9 References ...... 238

37 Shear mapping 239 37.1 Definition ...... 240 37.1.1 Horizontal and vertical shear of the plane ...... 240 37.1.2 General shear mappings ...... 241 37.2 Applications ...... 242 37.3 References ...... 242

38 Similarity (geometry) 243 38.1 Similar triangles ...... 244 xiv CONTENTS

38.2 Other similar polygons ...... 245 38.3 Similar curves ...... 245 38.4 Similarity in Euclidean space ...... 245 38.5 Ratios of sides, of areas, and of volumes ...... 246 38.6 Similarity in general metric spaces ...... 246 38.7 Topology ...... 247 38.8 Self-similarity ...... 247 38.9 See also ...... 248 38.10Notes ...... 248 38.11References ...... 249 38.12Further reading ...... 249 38.13External links ...... 249

39 Squeeze mapping 250 39.1 Logarithm and hyperbolic angle ...... 251 39.2 Group theory ...... 251 39.3 Applications ...... 252 39.3.1 Corner flow ...... 252 39.3.2 Relativistic spacetime ...... 253 39.3.3 Bridge to transcendentals ...... 253 39.4 See also ...... 253 39.5 References ...... 254

40 Transformation matrix 255 40.1 Uses ...... 255 40.2 Finding the matrix of a transformation ...... 255 40.2.1 Eigenbasis and diagonal matrix ...... 256 40.3 Examples in 2D computer graphics ...... 256 40.3.1 Rotation ...... 256 40.3.2 Shearing ...... 257 40.3.3 Reflection ...... 257 40.3.4 Orthogonal projection ...... 257 40.4 Examples in 3D computer graphics ...... 257 40.4.1 Rotation ...... 257 40.4.2 Reflection ...... 258 40.5 Composing and inverting transformations ...... 258 40.6 Other kinds of transformations ...... 258 40.6.1 Affine transformations ...... 258 40.6.2 Perspective projection ...... 261 40.7 See also ...... 261 40.8 References ...... 261 40.9 External links ...... 262 CONTENTS xv

41 Translation (geometry) 263 41.1 Matrix representation ...... 265 41.2 Translations in physics ...... 265 41.3 See also ...... 266 41.4 External links ...... 266 41.5 References ...... 266

42 Vector space 267 42.1 Introduction and definition ...... 268 42.1.1 First example: arrows in the plane ...... 268 42.1.2 Second example: ordered pairs of numbers ...... 268 42.1.3 Definition ...... 268 42.1.4 Alternative formulations and elementary consequences ...... 269 42.2 History ...... 269 42.3 Examples ...... 270 42.3.1 Coordinate spaces ...... 270 42.3.2 Complex numbers and other field extensions ...... 270 42.3.3 Function spaces ...... 271 42.3.4 Linear equations ...... 271 42.4 Basis and dimension ...... 271 42.5 Linear maps and matrices ...... 273 42.5.1 Matrices ...... 273 42.5.2 Eigenvalues and eigenvectors ...... 274 42.6 Basic constructions ...... 275 42.6.1 Subspaces and quotient spaces ...... 275 42.6.2 Direct product and direct sum ...... 277 42.6.3 Tensor product ...... 277 42.7 Vector spaces with additional structure ...... 278 42.7.1 Normed vector spaces and inner product spaces ...... 279 42.7.2 Topological vector spaces ...... 279 42.7.3 Algebras over fields ...... 282 42.8 Applications ...... 283 42.8.1 Distributions ...... 284 42.8.2 Fourier analysis ...... 284 42.8.3 Differential geometry ...... 285 42.9 Generalizations ...... 285 42.9.1 Vector bundles ...... 285 42.9.2 Modules ...... 286 42.9.3 Affine and projective spaces ...... 287 42.10See also ...... 288 42.11Notes ...... 288 42.12Footnotes ...... 289 xvi CONTENTS

42.13References ...... 292 42.13.1 Algebra ...... 292 42.13.2 Analysis ...... 292 42.13.3 Historical references ...... 293 42.13.4 Further references ...... 294 42.14External links ...... 295 42.15Text and image sources, contributors, and licenses ...... 296 42.15.1 Text ...... 296 42.15.2 Images ...... 303 42.15.3 Content license ...... 308 Chapter 1

2 × 2 real matrices

In mathematics, the set of 2×2 real matrices is denoted by M(2, R). Two matrices p and q in M(2, R) have a sum p + q given by matrix addition. The product matrix p q is formed from the dot product of the rows and columns of its factors through matrix multiplication. For

( ) a b q = , c d let

( ) d −b q∗ = . −c a Then q q* = q* q = (ad − bc) I, where I is the 2×2 identity matrix. The real number ad − bc is called the determinant of q. When ad − bc ≠ 0, q is an invertible matrix, and then q−1 = q∗ /(ad − bc). The collection of all such invertible matrices constitutes the general linear group GL(2, R). In terms of abstract algebra, M(2, R) with the associated addition and multiplication operations forms a ring, and GL(2, R) is its group of units. M(2, R) is also a four-dimensional vector space, so it is considered an associative algebra. It is ring-isomorphic to the coquaternions, but has a different profile. The 2×2 real matrices are in one-one correspondence with the linear mappings of the two-dimensional Cartesian coordinate system into itself by the rule

( ) ( )( ) ( ) x a b x ax + by 7→ = . y c d y cx + dy

1.1 Profile

Within M(2, R), the multiples by real numbers of the identity matrix I may be considered a real line. This real line is the place where all commutative subrings come together: Let Pm = {xI + ym : x, y ∈ R} where m2 ∈ { −I, 0, I }. Then Pm is a commutative subring and M(2, R) = ∪Pm where the union is over all m such that m2 ∈ { −I, 0, I }. To identify such m, first square the generic matrix:

( ) aa + bc ab + bd . ac + cd bc + dd

1 2 CHAPTER 1. 2 × 2 REAL MATRICES

When a + d = 0 this square is a diagonal matrix. Thus one assumes d = −a when looking for m to form commutative subrings. When mm = −I, then bc = −1 − aa, an equation describing a hyperbolic paraboloid in the space of parameters (a, b, c). Such an m serves as an imaginary unit. In this case Pm is isomorphic to the field of (ordinary) complex numbers. When mm = +I, m is an involutory matrix. Then bc = +1 − aa, also giving a hyperbolic paraboloid. If a matrix is an idempotent matrix, it must lie in such a Pm and in this case Pm is isomorphic to the ring of split-complex numbers. The case of a nilpotent matrix, mm = 0, arises when only one of b or c is non-zero, and the commutative subring Pm is then a copy of the dual number plane. When M(2, R) is reconfigured with a change of basis, this profile changes to the profile of split-quaternions where the sets of square roots of I and −I take a symmetrical shape as hyperboloids.

1.2 Equi-areal mapping

Main article: Equiareal map

First transform one differential vector into another:

( ) ( )( ) ( ) du p r dx p dx + r dy = = . dv q s dy q dx + s dy

Areas are measured with density dx ∧ dy , a differential 2-form which involves the use of exterior algebra. The transformed density is

du ∧ dv = 0 + ps dx ∧ dy + qr dy ∧ dx + 0 = (ps − qr) dx ∧ dy = (det g) dx ∧ dy.

Thus the equi-areal mappings are identified with SL(2, R) = {g ∈ M(2, R) : det(g) = 1}, the special linear group. Given the profile above, every such g lies in a commutative subring Pm representing a type of complex plane according to the square of m. Since g g* = I, one of the following three alternatives occurs:

• mm = −I and g is on a circle of Euclidean rotations; or

• mm = I and g is on an hyperbola of squeeze mappings; or

• mm = 0 and g is on a line of shear mappings.

Writing about planar affine mapping, Rafael Artzy made a similar trichotomy of planar, linear mapping in his book Linear Geometry (1965).

1.3 Functions of 2 × 2 real matrices

The commutative subrings of M(2, R) determine the function theory; in particular the three types of subplanes have their own algebraic structures which set the value of algebraic expressions. Consideration of the square root function and the logarithm function serves to illustrate the constraints implied by the special properties of each type of subplane Pm described in the above profile. The concept of identity component of the group of units of Pm leads to the polar decomposition of elements of the group of units:

• If mm = −I, then z = ρ exp(θm).

• If mm = 0, then z = ρ exp(s m) or z = − ρ exp(s m).

• If mm = I, then z = ρ exp(a m) or z = −ρ exp(a m) or z = m ρ exp(a m) or z = −m ρ exp(a m). 1.4. 2 × 2 REAL MATRICES AS COMPLEX NUMBERS 3

In the first case exp(θ m) = cos(θ) + m sin(θ). In the case of the dual numbers exp(s m) = 1 + s m. Finally, in the case of split complex numbers there are four components in the group of units. The identity component is parameterized by ρ and exp(a m) = cosh a + m sinh a. √ √ Now ρ exp(am) = ρ exp(am/2) regardless of the subplane Pm, but the argument of the function must be taken from the identity component of its group of units. Half the plane is lost in the case of the dual number structure; three-quarters of the plane must be excluded in the case of the split-complex number structure. Similarly, if ρ exp(a m) is an element of the identity component of the group of units of a plane associated with 2×2 matrix m, then the logarithm function results in a value log ρ + a m. The domain of the logarithm function suffers the same constraints as does the square root function described above: half or three-quarters of Pm must be excluded in the cases mm = 0 or mm = I. Further function theory can be seen in the article complex functions for the C structure, or in the article motor variable for the split-complex structure.

1.4 2 × 2 real matrices as complex numbers

Every 2×2 real matrix can be interpreted as one of three types of (generalized[1]) complex numbers: standard complex numbers, dual numbers, and split-complex numbers. Above, the algebra of 2×2 matrices is profiled as a union of complex planes, all sharing the same real axis. These planes are presented as commutative subrings Pm. We can determine to which complex plane a given 2×2 matrix belongs as follows and classify which kind of complex number that plane represents. Consider the 2×2 matrix

( ) a b z = . c d We seek the complex plane Pm containing z. As noted above, the square of the matrix z is diagonal when a + d = 0. The matrix z must be expressed as the sum of a multiple of the identity matrix I and a matrix in the hyperplane a + d = 0. Projecting z alternately onto these subspaces of R4 yields

a + d z = xI + n, x = , n = z − xI. 2 Furthermore,

2 2 (a−d) n = pI where p = 4 + bc .

Now z is one of three types of complex number:

• If p < 0, then it is an ordinary complex number: √ √ Let q = 1/ −p, m = qn . Then m2 = −I, z = xI + m −p .

• If p = 0, then it is the dual number:

z = xI + n

• If p > 0, then z is a split-complex number: √ √ Let q = 1/ p, m = qn . Then m2 = +I, z = xI + m p .

Similarly, a 2×2 matrix can also be expressed in polar coordinates with the caveat that there are two connected components of the group of units in the dual number plane, and four components in the split-complex number plane. 4 CHAPTER 1. 2 × 2 REAL MATRICES

1.5 References

[1] Anthony A. Harkin & Joseph B. Harkin (2004) Geometry of Generalized Complex Numbers, Mathematics Magazine 77(2):118–29

• Rafael Artzy (1965) Linear Geometry, Chapter 2-6 Subgroups of the Plane Affine Group over the Real Field, p. 94, Addison-Wesley.

• Helmut Karzel & Gunter Kist (1985) “Kinematic Algebras and their Geometries”, found in • Rings and Geometry, R. Kaya, P. Plaumann, and K. Strambach editors, pp. 437–509, esp 449,50, D. Reidel ISBN 90-277-2112-2 . • Svetlana Katok (1992) Fuchsian groups, pp. 113ff, University of Chicago Press ISBN 0-226-42582-7 .

• Garret Sobczyk (2012). “Chapter 2: Complex and Hyperbolic Numbers”. New Foundations in Mathematics: The Geometric Concept of Number. Birkhäuser. ISBN 978-0-8176-8384-9. Chapter 2

3D projection

3D projection is any method of mapping three-dimensional points to a two-dimensional plane. As most current methods for displaying graphical data are based on planar (pixel information from several bitplanes) two-dimensional media, the use of this type of projection is widespread, especially in computer graphics, engineering and drafting.

2.1 Orthographic projection

Main article: Orthographic projection

When the human eye looks at a scene, objects in the distance appear smaller than objects close by. Orthographic projection ignores this effect to allow the creation of to-scale drawings for construction and engineering. Orthographic projections are a small set of transforms often used to show profile, detail or precise measurements of a three dimensional object. Common names for orthographic projections include plane, cross-section, bird’s-eye, and elevation. If the normal of the viewing plane (the camera direction) is parallel to one of the primary axes (which is the x, y, or z axis), the mathematical transformation is as follows; To project the 3D point ax , ay , az onto the 2D point bx , by using an orthographic projection parallel to the y axis (profile view), the following equations can be used:

bx = sxax + cx by = szaz + cz where the vector s is an arbitrary scale factor, and c is an arbitrary offset. These constants are optional, and can be used to properly align the viewport. Using matrix multiplication, the equations become:

  [ ] [ ] [ ] ax bx sx 0 0   cx = ay + by 0 0 sz cz az While orthographically projected images represent the three dimensional nature of the object projected, they do not represent the object as it would be recorded photographically or perceived by a viewer observing it directly. In particular, parallel lengths at all points in an orthographically projected image are of the same scale regardless of whether they are far away or near to the virtual viewer. As a result, lengths near to the viewer are not foreshortened as they would be in a perspective projection.

2.2 Weak perspective projection

A “weak” perspective projection uses the same principles of an orthographic projection, but requires the scaling factor to be specified, thus ensuring that closer objects appear bigger in the projection, and vice versa. It can be seen as a

5 6 CHAPTER 2. 3D PROJECTION

hybrid between an orthographic and a perspective projection, and described either as a perspective projection with [1] individual point depths Zi replaced by an average constant depth Zave , or simply as an orthographic projection plus a scaling.[2] The weak-perspective model thus approximates perspective projection while using a simpler model, similar to the pure (unscaled) orthographic perspective. It is a reasonable approximation when the depth of the object along the line of sight is small compared to the distance from the camera, and the field of view is small. With these conditions, it can be assumed that all points on a 3D object are at the same distance Zave from the camera without significant errors in the projection (compared to the full perspective model).

2.3 Perspective projection

See also: Transformation matrix and Camera matrix

When the human eye views a scene, objects in the distance appear smaller than objects close by - this is known as perspective. While orthographic projection ignores this effect to allow accurate measurements, perspective projection shows distant objects as smaller to provide additional realism. The perspective projection requires a more involved definition as compared to orthographic projections. A conceptual aid to understanding the mechanics of this projection is to imagine the 2D projection as though the object(s) are being viewed through a camera viewfinder. The camera’s position, orientation, and field of view control the behavior of the projection transformation. The following variables are defined to describe this transformation:

• ax,y,z - the 3D position of a point A that is to be projected.

• cx,y,z - the 3D position of a point C representing the camera.

• θx,y,z - The orientation of the camera (represented by Tait–Bryan angles). [3] • ex,y,z - the viewer’s position relative to the display surface which goes through point C representing the camera.

Which results in:

• bx,y - the 2D projection of a .

When cx,y,z = ⟨0, 0, 0⟩, and θx,y,z = ⟨0, 0, 0⟩, the 3D vector ⟨1, 2, 0⟩ is projected to the 2D vector ⟨1, 2⟩ .

Otherwise, to compute bx,y we first define a vector dx,y,z as the position of point A with respect to a coordinate system defined by the camera, with origin in C and rotated by θ with respect to the initial coordinate system. This is achieved by subtracting c from a and then applying a rotation by −θ to the result. This transformation is often called a camera transform, and can be expressed as follows, expressing the rotation in terms of rotations about the x, y, and z axes (these calculations assume that the axes are ordered as a left-handed system of axes): [4] [5]

          dx 1 0 0 cos(−θy) 0 sin(−θy) cos(−θz) − sin(−θz) 0 ax cx           dy = 0 cos(−θx) − sin(−θx) 0 1 0 sin(−θz) cos(−θz) 0 ay − cy dz 0 sin(−θx) cos(−θx) − sin(−θy) 0 cos(−θy) 0 0 1 az cz This representation corresponds to rotating by three Euler angles (more properly, Tait–Bryan angles), using the xyz convention, which can be interpreted either as “rotate about the extrinsic axes (axes of the scene) in the order z, y, x (reading right-to-left)" or “rotate about the intrinsic axes (axes of the camera) in the order x, y, z (reading left-to- right)". Note that if the camera is not rotated ( θx,y,z = ⟨0, 0, 0⟩ ), then the matrices drop out (as identities), and this reduces to simply a shift: d = a − c. Alternatively, without using matrices (let’s replace (aₓ-cₓ) with x and so on, and abbreviate cosθ to c and sinθ to s):

dx = cy(szy + czx) − syz dy = sx(cyz + sy(szy + czx)) + cx(czy − szx) dz = cx(cyz + sy(szy + czx)) − sx(czy − szx) 2.3. PERSPECTIVE PROJECTION 7

This transformed point can then be projected onto the 2D plane using the formula (here, x/y is used as the projection plane; literature also may use x/z):[6]

ez bx = dx − ex dz ez . by = dy − ey dz

Or, in matrix form using homogeneous coordinates, the system

     f 1 0 − ex 0 d x ez x    ey   fy  0 1 − 0dy   =  ez   fz 0 0 1 0 dz fw 0 0 1/ez 0 1

in conjunction with an argument using similar triangles, leads to division by the homogeneous coordinate, giving

b = f /f x x w . by = fy/fw

−1 The distance of the viewer from the display surface, ez , directly relates to the field of view, where α = 2·tan (1/ez) is the viewed angle. (Note: This assumes that you map the points (−1,−1) and (1,1) to the corners of your viewing surface) The above equations can also be rewritten as:

b = (d s )/(d r )r x x x z x z . by = (dysy)/(dzry)rz

In which sx,y is the display size, rx,y is the recording surface size (CCD or film), rz is the distance from the recording surface to the entrance pupil (camera center), and dz is the distance, from the 3D point being projected, to the entrance pupil. Subsequent clipping and scaling operations may be necessary to map the 2D plane onto any particular display media. 8 CHAPTER 2. 3D PROJECTION

2.4 Diagram

Bz Az

Bx

Ax

To determine which screen x-coordinate corresponds to a point at Ax,Az multiply the point coordinates by:

Bz Bx = Ax Az where

Bx is the screen x coordinate

Ax is the model x coordinate

Bz is the focal length—the axial distance from the camera center to the image plane

Az is the subject distance.

Because the camera is in 3D, the same works for the screen y-coordinate, substituting y for x in the above diagram and equation.

2.5 See also

• 3D computer graphics • Camera matrix • Computer graphics • Graphics card • Homography • Homogeneous coordinates • Perspective (graphical) 2.6. REFERENCES 9

• Texture mapping

• Virtual globe • Transform and lighting

2.6 References

[1] Subhashis Banerjee (2002-02-18). “The Weak-Perspective Camera”.

[2] Alter, T. D. (July 1992). 3D Pose from 3 Corresponding Points under Weak-Perspective Projection (PDF) (Technical report). MIT AI Lab.

[3] Ingrid Carlbom, Joseph Paciorek (1978). “Planar Geometric Projections and Viewing Transformations” (PDF). ACM Computing Surveys 10 (4): 465–502. doi:10.1145/356744.356750..

[4] Riley, K F (2006). Mathematical Methods for Physics and Engineering. Cambridge University Press. pp. 931, 942. doi:10.2277/0521679710. ISBN 0-521-67971-0.

[5] Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, Mass.: Addison-Wesley Pub. Co. pp. 146–148. ISBN 0-201-02918-9.

[6] Sonka, M; Hlavac, V; Boyle, R (1995). Image Processing, Analysis & Machine Vision (2nd ed.). Chapman and Hall. p. 14. ISBN 0-412-45570-6.

2.7 External links

• A case study in camera projection

• Creating 3D Environments from Digital Photographs

2.8 Further reading

• Kenneth C. Finney (2004). 3D Game Programming All in One. Thomson Course. p. 93. ISBN 978-1-59200- 136-1. • Koehler; Dr. Ralph. 2D/3D Graphics and Splines with Source Code. ISBN 0759611874. Chapter 3

Affine coordinate system

This article is about affine coordinates in affine geometry. For affine coordinate systems in algebraic geometry, see Algebraic variety.

In mathematics, an affine coordinate system is a coordinate system on an affine space where each coordinate is an affine map to the number line. In other words, it is an injective affine map from an affine space A to the coordinate space Kn, where K is the field of scalars, for example, the real numbers R. The most important case of affine coordinates in Euclidean spaces is real-valued Cartesian coordinate system. Orthogonal affine coordinate systems are rectangular, and others are referred to as oblique.

A system of n coordinates on n-dimensional space is defined by a (n+1)-tuple (O, R1,… Rn) of points not belonging to any affine subspace of a lesser dimension. Any given coordinate n-tuple gives the point by the formula:

(x1,… xn) ↦ O + x1 (R1 − O) + … + xn (Rn − O).

Note that Rj − O are difference vectors with the origin in O and ends in R . An affine space cannot have a coordinate system with n less than its dimension, but n may indeed be greater, which means that the coordinate map is not necessary surjective. Examples of n-coordinate system in an (n−1)-dimensional space are barycentric coordinates and affine “homogeneous” coordinates (1, x1,…, xn₋₁). In the latter case the x0 coordinate is equal to 1 on all space, but this “reserved” coordinate allows for matrix representation of affine maps similar to one used for projective maps.

3.1 See also

• Convex combination • Centroid, can be calculated in affine coordinates

• Homogeneous coordinates, a similar concept but without uniqueness of values

10 Chapter 4

Affine geometry

A1 A2

B1 B2

C1 C2

In affine geometry, one uses Playfair’s axiom to find the line through C1 and parallel to B1B2, and to find the line through B2 and parallel to B1C1: their intersection C2 is the result of the indicated translation.

In mathematics, affine geometry is what remains of Euclidean geometry, when not using (mathematicians often say “when forgetting”) the metric notions of distance and angle. As the notion of parallel lines is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines. Therefore, Playfair’s axiom (given a line L and a point P not on L, there is exactly one line parallel to L that passes through P) is fundamental in affine geometry. Comparisons of figures in affine geometry are made with affine transformations, which are mappings that preserve alignment of points and parallelism of lines. Affine geometry can be developed in two ways that are essentially equivalent.[1] In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair’s axiom). Affine geometry can also be developed on the basis of linear algebra. In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space

11 12 CHAPTER 4. AFFINE GEOMETRY

(over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations. In more concrete terms, this amounts to having an operation that associates to any ordered pair of points a vector and another operation that allows translation of a point by a vector to give another point; these operations are required to satisfy a number of axioms (notably that two successive translations have the effect of translation by the sum vector). By choosing any point as “origin”, the points are in one-to-one correspondence with the vectors, but there is no preferred choice for the origin; thus an affine space may be viewed as obtained from its associated vector space by “forgetting” the origin (zero vector).

4.1 History

In 1748, Euler introduced the term affine[2][3] (Latin affinis, “related”) in his book Introductio in analysin infinitorum (volume 2, chapter XVIII). In 1827, August Möbius wrote on affine geometry in his Der barycentrische Calcul (chapter 3). After Felix Klein's Erlangen program, affine geometry was recognized as a generalization of Euclidean geometry.[4] In 1912, Edwin B. Wilson and Gilbert N. Lewis developed an affine geometry[5][6] to express the special theory of relativity. In 1918, Hermann Weyl referred to affine geometry for his text Space, Time, Matter. He used affine geometry to introduce vector addition and subtraction[7] at the earliest stages of his development of mathematical physics. Later, E. T. Whittaker wrote:[8]

Weyl’s geometry is interesting historically as having been the first of the affine geometries to be worked out in detail: it is based on a special type of parallel transport [...using] worldlines of light-signals in four-dimensional space-time. A short element of one of these world-lines may be called a null-vector; then the parallel transport in question is such that it carries any null-vector at one point into the position of a null-vector at a neighboring point.

In 1984, “the affine plane associated to the Lorentzian vector space L2 " was described by Graciela Birman and Katsumi Nomizu in an article entitled “Trigonometry in Lorentzian geometry”.[9]

4.2 Systems of axioms

Several axiomatic approaches to affine geometry have been put forward:

4.2.1 Pappus’ law

As affine geometry deals with parallel lines, one of the properties of parallels noted by Pappus of Alexandria has been taken as a premise:[10][11]

• If A, B, C are on one line and A′,B′,C′ on another, then

(AB′ ∥ A′B ∧ BC′ ∥ B′C) ⇒ CA′ ∥ C′A. The full axiom system proposed has point, line, and line containing point as primitive notions:

• Two points are contained in just one line.

• For any line l and any point P, not on l, there is just one line containing P and not containing any point of l. This line is said to be parallel to l.

• Every line contains at least two points.

• There are at least three points not belonging to one line. 4.2. SYSTEMS OF AXIOMS 13

C1

B1

A1

O

C2 B2 A2

Pappus law: if the red lines are parallel and the blue lines are parallel, then the dotted black lines must be parallel.

According to H. S. M. Coxeter:

The interest of these five axioms is enhanced by the fact that they can be developed into a vast body of propositions, holding not only in Euclidean geometry but also in Minkowski’s geometry of time and space (in the simple case of 1 + 1 dimensions, whereas the special theory of relativity needs 1 + 3). The extension to either Euclidean or Minkowskian geometry is achieved by adding various further axioms of orthogonality, etc[12]

The various types of affine geometry correspond to what interpretation is taken for rotation. Euclidean geometry corresponds to the ordinary idea of rotation, while Minkowski’s geometry corresponds to hyperbolic rotation. With respect to perpendicular lines, they remain perpendicular when the plane is subjected to ordinary rotation. In the Minkowski geometry, lines that are hyperbolic-orthogonal remain in that relation when the plane is subjected to hyperbolic rotation.

4.2.2 Ordered structure

An axiomatic treatment of plane affine geometry can be built from the axioms of ordered geometry by the addition of two additional axioms:[13]

1. (Affine axiom of parallelism) Given a point A and a line r, not through A, there is at most one line through A which does not meet r. 2. (Desargues) Given seven distinct points A, A', B, B', C, C', O, such that AA', BB', and CC' are distinct lines through O and AB is parallel to A'B' and BC is parallel to B'C', then AC is parallel to A'C'.

The affine concept of parallelism forms an equivalence relation on lines. Since the axioms of ordered geometry as presented here include properties that imply the structure of the real numbers, those properties carry over here so that this is an axiomatization of affine geometry over the field of real numbers.

4.2.3 Ternary rings

Main article: planar ternary ring 14 CHAPTER 4. AFFINE GEOMETRY

The first non-Desarguesian plane was noted by David Hilbert in his Foundations of Geometry.[14] The Moulton plane is a standard illustration. In order to provide a context for such geometry as well as those where Desargues theorem is valid, the concept of a ternary ring has been developed. Rudimentary affine planes are constructed from ordered pairs taken from a ternary ring. A plane is said to have the “minor affine Desargues property” when two triangles in parallel perspective, having two parallel sides, must also have the third sides parallel. If this property holds in the rudimentary affine plane defined by a ternary ring, then there is an equivalence relation between “vectors” defined by pairs of points from the plane.[15] Furthermore, the vectors form an abelian group under addition, the ternary ring is linear, and satisfies right distributivity:

(a + b) c = ac + bc.

4.3 Affine transformations

Main article: Affine transformation

Geometrically, affine transformations (affinities) preserve collinearity: so they transform parallel lines into parallel lines and preserve ratios of distances along parallel lines. We identify as affine theorems any geometric result that is invariant under the affine group (in Felix Klein's Erlangen programme this is its underlying group of symmetry transformations for affine geometry). Consider in a vector space V, the general linear group GL(V). It is not the whole affine group because we must allow also translations by vectors v in V. (Such a translation maps any w in V to w + v.) The affine group is generated by the general linear group and the translations and is in fact their semidirect product V ⋊ GL(V ) . (Here we think of V as a group under its operation of addition, and use the defining representation of GL(V) on V to define the semidirect product.) For example, the theorem from the plane geometry of triangles about the concurrence of the lines joining each vertex to the midpoint of the opposite side (at the centroid or barycenter) depends on the notions of mid-point and centroid as affine invariants. Other examples include the theorems of Ceva and Menelaus. Affine invariants can also assist calculations. For example, the lines that divide the area of a triangle into two equal halves form an envelope inside the triangle. The ratio of the area of the envelope to the area of the triangle is affine invariant, and so only needs to be calculated from a simple case such as a unit isosceles right angled triangle to give 3 − 1 4 loge(2) 2 , i.e. 0.019860... or less than 2%, for all triangles. Familiar formulas such as half the base times the height for the area of a triangle, or a third the base times the height for the volume of a pyramid, are likewise affine invariants. While the latter is less obvious than the former for the general case, it is easily seen for the one-sixth of the unit cube formed by a face (area 1) and the midpoint of the cube (height 1/2). Hence it holds for all pyramids, even slanting ones whose apex is not directly above the center of the base, and those with base a parallelogram instead of a square. The formula further generalizes to pyramids whose base can be dissected into parallelograms, including cones by allowing infinitely many parallelograms (with due attention to convergence). The same approach shows that a four-dimensional pyramid has 4D volume one quarter the 3D volume of its parallelepiped base times the height, and so on for higher dimensions.

4.4 Affine space

Main article: Affine space

Affine geometry can be viewed as the geometry of an affine space of a given dimension n, coordinatized over a field K. There is also (in two dimensions) a combinatorial generalization of coordinatized affine space, as developed in synthetic finite geometry. In projective geometry, affine space means the complement of a hyperplane at infinity in a projective space. Affine space can also be viewed as a vector space whose operations are limited to those linear combinations whose coefficients sum to one, for example 2x − y, x − y + z,(x + y + z)/3, ix + (1 − i)y, etc. Synthetically, affine planes are 2-dimensional affine geometries defined in terms of the relations between points and lines (or sometimes, in higher dimensions, hyperplanes). Defining affine (and projective) geometries as configurations of points and lines (or hyperplanes) instead of using coordinates, one gets examples with no coordinate fields. A major 4.5. PROJECTIVE VIEW 15 property is that all such examples have dimension 2. Finite examples in dimension 2 (finite affine planes) have been valuable in the study of configurations in infinite affine spaces, in group theory, and in combinatorics. Despite being less general than the configurational approach, the other approaches discussed have been very successful in illuminating the parts of geometry that are related to symmetry.

4.5 Projective view

In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry. On the one hand, affine geometry is Euclidean geometry with congruence left out; on the other hand, affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the points at infinity.[16] In affine geometry, there is no metric structure but the parallel postulate does hold. Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality.[17] In this viewpoint, an affine transformation geometry is a group of projective transformations that do not permute finite points with points at infinity.

4.6 See also

• Non-Euclidean geometry

4.7 References

[1] Artin, Emil (1988), Geometric algebra, Wiley Classics Library, New York: John Wiley & Sons Inc., pp. x+214, ISBN 0-471-60839-4, MR 1009557 (Reprint of the 1957 original; A Wiley-Interscience Publication)

[2] Miller, Jeff. “Earliest Known Uses of Some of the Words of Mathematics (A)".

[3] Blaschke, Wilhelm (1954). Analytische Geometrie. Basel: Birkhauser. p. 31.

[4] Coxeter, H. S. M. (1969). Introduction to Geometry. New York: John Wiley & Sons. p. 191. ISBN 0-471-50458-0.

[5] Edwin B. Wilson & Gilbert N. Lewis (1912). “The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics”, Proceedings of the American Academy of Arts and Sciences 48:387–507

[6] Synthetic Spacetime, a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by WebCite

[7] Hermann Weyl (1918)Raum, Zeit, Materie. 5 edns. to 1922 ed. with notes by Jūrgen Ehlers, 1980. trans. 4th edn. Henry Brose, 1922 Space Time Matter, Methuen, rept. 1952 Dover. ISBN 0-486-60267-2 . See Chapter 1 §2 Foundations of Affine Geometry, pp 16–27

[8] E. T. Whittaker (1958). From Euclid to Eddington: a study of conceptions of the external world, Dover Publications, p. 130.

[9] Graciela S. Birman & Katsumi Nomizu (1984). “Trigonometry in Lorentzian geometry”, American Mathematical Monthly 91(9):543–9, Lorentzian affine plane: p. 544

[10] Veblen 1918: p. 103 (figure), and p. 118 (exercise 3).

[11] Coxeter 1955, The Affine Plane, § 2: Affine geometry as an independent system

[12] Coxeter 1955, Affine plane, p. 8

[13] Coxeter, Introduction to Geometry, p. 192

[14] David Hilbert, 1980 (1899). The Foundations of Geometry, 2nd ed., Chicago: Open Court, weblink from Project Guten- berg, p. 74.

[15] Rafael Artzy (1965). Linear Geometry, Addison-Wesley, p. 213.

[16] H. S. M. Coxeter (1942). Non-Euclidean Geometry, University of Toronto Press, pp. 18, 19.

[17] Coxeter 1942, p. 178 16 CHAPTER 4. AFFINE GEOMETRY

4.8 Further reading

• Emil Artin (1957) Geometric Algebra, chapter 2: “Affine and projective geometry”, Interscience Publishers.

• V.G. Ashkinuse & Isaak Yaglom (1962) Ideas and Methods of Affine and Projective Geometry (in Russian), Ministry of Education, Moscow.

• M. K. Bennett (1995) Affine and Projective Geometry, John Wiley & Sons ISBN 0-471-11315-8 .

• H. S. M. Coxeter (1955) “The Affine Plane”, Scripta Mathematica 21:5–14, a lecture delivered before the Forum of the Society of Friends of Scripta Mathematica on Monday, April 26, 1954.

• Felix Klein (1939) Elementary Mathematics from an Advanced Standpoint: Geometry, translated by E. R. Hedrick and C. A. Noble, pp 70–86, Macmillan Company.

• Bruce E. Meserve (1955) Fundamental Concepts of Geometry, Chapter 5 Affine Geometry,, pp 150–84, Addison-Wesley.

• Peter Scherk & Rolf Lingenberg (1975) Rudiments of Plane Affine Geometry, Mathematical Expositions #20, University of Toronto Press.

• Wanda Szmielew (1984) From Affine to Euclidean Geometry: an axiomatic approach, D. Reidel, ISBN 90- 277-1243-3 .

• Oswald Veblen (1918) Projective Geometry, volume 2, chapter 3: Affine group in the plane, pp 70 to 118, Ginn & Company.

4.9 External links

• Peter Cameron's Projective and Affine Geometries from University of London. • Jean H. Gallier (2001). Geometric Methods and Applications for Computer Science and Engineering, Chapter 2: “Basics of Affine Geometry” (PDF), Springer Texts in Applied Mathematics #38, chapter online from University of Pennsylvania. Chapter 5

Affine group

In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself. It is a Lie group if K is the real or complex field or quaternions.

5.1 Relation to general linear group

5.1.1 Construction from general linear group

Concretely, given a vector space V, it has an underlying affine space A obtained by “forgetting” the origin, with V acting by translations, and the affine group of A can be described concretely as the semidirect product of V by GL(V), the general linear group of V:

Aff(A) = V ⋊ GL(V )

The action of GL(V) on V is the natural one (linear transformations are automorphisms), so this defines a semidirect product. In terms of matrices, one writes:

Aff(n, K) = Kn ⋊ GL(n, K)

where here the natural action of GL(n,K) on Kn is matrix multiplication of a vector.

5.1.2 Stabilizer of a point

Given the affine group of an affine space A, the stabilizer of a point p is isomorphic to the general linear group of the same dimension (so the stabilizer of a point in Aff(2,R) is isomorphic to GL(2,R)); formally, it is the general linear group of the vector space (A, p) : recall that if one fixes a point, an affine space becomes a vector space. All these subgroups are conjugate, where conjugation is given by translation from p to q (which is uniquely defined), however, no particular subgroup is a natural choice, since no point is special – this corresponds to the multiple choices of transverse subgroup, or splitting of the short exact sequence

1 → V → V ⋊ GL(V ) → GL(V ) → 1

In the case that the affine group was constructed by starting with a vector space, the subgroup that stabilizes the origin (of the vector space) is the original GL(V).

17 18 CHAPTER 5. AFFINE GROUP

5.2 Matrix representation

Representing the affine group as a semidirect product of V by GL(V), then by construction of the semidirect product, the elements are pairs (M, v), where v is a vector in V and M is a linear transform in GL(V), and multiplication is given by:

(M, v) · (N, w) = (MN, v + Mw).

This can be represented as the (n + 1)×(n + 1) block matrix:

( ) M v 0 1

where M is an n×n matrix over K, v an n × 1 column vector, 0 is a 1 × n row of zeros, and 1 is the 1 × 1 identity block matrix. Formally, Aff(V) is naturally isomorphic to a subgroup of GL(V ⊕ K) , with V embedded as the affine plane {(v, 1)|v ∈ V } , namely the stabilizer of this affine plane; the above matrix formulation is the (transpose of) the realization of this, with the (n × n and 1 × 1) blocks corresponding to the direct sum decomposition V ⊕ K . A similar representation is any (n + 1)×(n + 1) matrix in which the entries in each column sum to 1.[1] The similarity P for passing from the above kind to this kind is the (n + 1)×(n + 1) identity matrix with the bottom row replaced by a row of all ones. Each of these two classes of matrices is closed under matrix multiplication.

5.3 Planar affine group

According to Artzy,[2] “The linear part of each affinity [of the real affine plane] can be brought into one of the following standard forms by a coordinate transformation followed by a dilation from the origin:

1. x 7→ ax + by, y 7→ −bx + ay, a, b ≠ 0, a2 + b2 = 1, 2. x 7→ x + by, y 7→ y, b ≠ 0, 3. x 7→ ax, y 7→ y/a, a ≠ 0, where the coefficients a, b, c, and d are real numbers.”

Case (1) corresponds to similarity transformations which generate a subgroup of similarities. Euclidean geometry corresponds to the subgroup of congruencies. It is characterized by Euclidean distance or angle, which are invariant under the subgroup of rotations. Case (2) corresponds to shear mappings. An important application is absolute time and space where Galilean trans- formations relate frames of reference. They generate the Galilean group. Case (3) corresponds to squeeze mapping. These transformations generate a subgroup, of the planar affine group, called the Lorentz group of the plane. The geometry associated with this group is characterized by hyperbolic angle, which is a measure that is invariant under the subgroup of squeeze mappings. Using the above matrix representation of the affine group on the plane, the matrix M is a 2 × 2 real matrix. Accord- ingly, a non-singular M must have one of three forms that correspond to the trichotomy of Artzy.

5.4 Other affine groups

5.4.1 General case

Given any subgroup G < GL(V ) of the general linear group, one can produce an affine group, sometimes denoted Aff(G) analogously as Aff(G) := V ⋊ G . 5.5. SEE ALSO 19

More generally and abstractly, given any group G and a representation of G on a vector space V, ρ: G → GL(V ) [3] one gets an associated affine group V ⋊ρ G : one can say that the affine group obtained is “a group extension by a vector representation”, and as above, one has the short exact sequence:

1 → V → V ⋊ρ G → G → 1.

5.4.2 Special affine group

Main article: Special affine group

The subset of all invertible affine transformations preserving a fixed volume form, or in terms of the semi-direct product, the set of all elements (M,v) with M of determinant 1, is a subgroup known as the special affine group.

5.4.3 Projective subgroup

Presuming knowledge of projectivity and the projective group of projective geometry, the affine group can be easily specified. For example, Günter Ewald wrote:.[4]

The set P of all projective collineations of Pn is a group which we may call the projective group of Pn. If we proceed from Pn to the affine space An by declaring a hyperplane ω to be a hyperplane at infinity, we obtain the affine group A of An as the subgroup of P consisting of all elements of P that leave ω fixed. A ⊂ P

5.4.4 Poincaré group

Main article: Poincaré group

The Poincaré group is the affine group of the Lorentz group O(1, 3) : R1,3 ⋊ O(1, 3) This example is very important in relativity.

5.5 See also

• Holomorph • Related articles on a diagram

5.6 References

[1] David G. Poole, “The Stochastic Group'", American Mathematical Monthly, volume 102, number 9 (November, 1995), pages 798–801 [2] Artzy p 94 [3] Since GL(V ) < Aut(V ) . Note that this containment is in general proper, since by “automorphisms” one means group au- tomorphisms, i.e., they preserve the group structure on V (the addition and origin), but not necessarily scalar multiplication, and these groups differ if working over R. [4] Günter Ewald (1971) Geometry: An Introduction, p. 241, Belmont: Wadsworth ISBN 9780534000349

• Rafael Artzy (1965) Linear Geometry, Chapter 2-6 Subgroups of the Plane Affine Group over the Real Field, Addison-Wesley. • Roger Lyndon (1985) Groups and Geometry, Section VI.1, Cambridge University Press, ISBN 0-521-31694-4. Chapter 6

Affine space

Not to be confused with affinity space. For a concept in algebraic geometry, see affine space (algebraic geometry). In mathematics, an affine space is a geometric structure that generalizes certain properties of parallel lines in

Line segments on a two-dimensional affine space

20 6.1. INFORMAL DESCRIPTIONS 21

Euclidean space. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, commonly called translation vector, between two points of the space. Thus it makes sense to subtract two points of the space, giving a vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a vector to a point of an affine space, resulting in a new point displaced from the starting point by that vector. An example of an affine space is a linear subspace of a vector space that has been translated away from the origin. In finite dimensions, such an affine subspace corresponds to the solution set of an inhomogeneous linear system. The displacement vectors for that affine space live in the solution set of the corresponding homogeneous linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space. The dimension of the vector space of the translations, is called the dimension of the affine space. An affine space of dimension one is an affine line. An affine space of dimension 2 is an affine plane. An affine subspace of dimension n – 1 in an affine space or a vector space of dimension n is a affine hyperplane.

6.1 Informal descriptions

The following characterization may be easier to understand than the usual formal definition: an affine space is what is left of a vector space after you've forgotten which point is the origin (or, in the words of the French mathematician Marcel Berger, “An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps”[1]). Imagine that Alice knows that a certain point is the actual origin, but Bob believes that another point — call it p — is the origin. Two vectors, a and b, are to be added. Bob draws an arrow from point p to point a and another arrow from point p to point b, and completes the parallelogram to find what Bob thinks is a + b, but Alice knows that he has actually computed

p + (a − p) + (b − p).

Similarly, Alice and Bob may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However, if the sum of the coefficients in a linear combination is 1, then Alice and Bob will arrive at the same answer. If Bob travels to

λa + (1 − λ)b

then Alice can similarly travel to

p + λ(a − p) + (1 − λ)(b − p) = λa + (1 − λ)b.

Then, for all coefficients λ + (1 − λ) = 1, Alice and Bob describe the same point with the same linear combination, starting from different origins. While Alice knows the “linear structure”, both Alice and Bob know the “affine structure”—i.e. the values of affine combinations, defined as linear combinations in which the sum of the coefficients is 1. An underlying set with an affine structure is an affine space.

6.2 Definition

An affine space[2] is a set A together with a vector space V over a field F and a transitive and free group action of V (with addition of vectors as group action) on A. (That is, an affine space is a principal homogeneous space for the action of V.) Explicitly, an affine space is a point set A together with a map

l : V × A → A, (v, a) 7→ v + a with the following properties:.[3][4][5] 22 CHAPTER 6. AFFINE SPACE

1. Left identity ∀a ∈ A, 0 + a = a

2. Associativity ∀v, w ∈ V, ∀a ∈ A, v + (w + a) = (v + w) + a

3. Uniqueness ∀a ∈ A, V → A: v 7→ v + a is a bijection.

(Since the group V is abelian, there is no difference between its left and right actions, so it is also permissible to place vectors on the right.) By choosing an origin, o, one can thus identify A with V, hence turn A into a vector space. Conversely, any vector space, V, is an affine space over itself.

6.2.1 Subtraction and Weyl’s axioms

The uniqueness property ensures that subtraction of any two elements of A is well defined, producing a vector of V:

a − b is the unique vector in V such that (a − b) + b = a .

One can equivalently define an affine space as a point set A, together with a vector space V, and a subtraction map

−→ ϕ : A × A → V, (a, b) 7→ b − a ≡ ab with the following properties:[6]

1. ∀p ∈ A, ∀v ∈ V there is a unique point q ∈ A such that q − p = v and 2. ∀p, q, r ∈ A, (q − p) + (r − q) = r − p .

These two properties are called Weyl's axioms.

6.2.2 Affine combinations

For any choice of origin o, and two points a, b in A and scalar λ, there is a unique element of A, denoted by λa + (1 − λ)b such that

(λa + (1 − λ)b) − o = λ(a − o) + (1 − λ)(b − o). This element can be shown to be independent of the choice of origin o. Instead of arbitrary linear combinations, only such affine combinations of points have meaning.

6.3 Examples

• When children find the answers to sums such as 4 + 3 or 4 − 2 by counting right or left on a number line, they are treating the number line as a one-dimensional affine space. • Any coset of a subspace V of a vector space is an affine space over that subspace. • If T is a matrix and b lies in its column space, the set of solutions of the equation T x = b is an affine space over the subspace of solutions of T x = 0. • The solutions of an inhomogeneous linear differential equation form an affine space over the solutions of the corresponding homogeneous linear equation. • Generalizing all of the above, if T : V → W is a linear mapping and y lies in its image, the set of solutions x ∈ V to the equation T x = y is a coset of the kernel of T , and is therefore an affine space over Ker T . 6.4. AFFINE SUBSPACES 23

6.4 Affine subspaces

An affine subspace (sometimes called a linear manifold, linear variety, or a flat) of a vector space V is a subset closed under affine combinations of vectors in the space. For example, the set

{ N N } ∑ ∑ A = αivi αi = 1 i i

is an affine space, where {vi}i∈I is a family of vectors in V; this space is the affine span of these points. To see that this is indeed an affine space, observe that this set carries a transitive action of the vector subspace W of V

{ N N } ∑ ∑ W = βivi βi = 0 . i i This affine subspace can be equivalently described as the coset of the W-action

S = p + W, where p is any element of A, or equivalently as any level set of the quotient map V → V/W. A choice of p gives a base point of A and an identification of W with A, but there is no natural choice, nor a natural identification of W with A. A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations. A linear subspace is an affine subspace containing the origin, or, equivalently, a subspace that is closed under linear combinations. For example, in R3 , the origin, lines and planes through the origin and the whole space are linear subspaces, while points, lines and planes in general as well as the whole space are the affine subspaces.

6.5 Affine combinations and affine dependence

Main article: Affine combination

An affine combination is a linear combination in which the sum of the coefficients is 1. Just as members of a set of vectors are linearly independent if none is a linear combination of the others, so also they are affinely independent if none is an affine combination of the others. The set of linear combinations of a set of vectors is their “linear span” and is always a linear subspace; the set of all affine combinations is their “affine span” and is always an affine subspace. For example, the affine span of a set of two points is the line that contains both; the affine span of a set of three non-collinear points is the plane that contains all three. Vectors

v1, v2,…, vn

are linearly dependent if there exist scalars a1, a2,…, an, not all zero, for which

Similarly they are affinely dependent if in addition the sum of coefficients is zero:

∑n ai = 0 i=1 a condition that ensures that the combination (1) makes sense as a displacement vector. 24 CHAPTER 6. AFFINE SPACE

6.6 Geometric objects as points and vectors

In an affine space, geometric objects have two different (although related) descriptions on languages of points (ele- ments of A) and vectors (elements of V ). A vector description can specify an object only up to translations.

6.7 Axioms

Affine space is usually studied as analytic geometry using coordinates, or equivalently vector spaces. It can also be studied as synthetic geometry by writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space. Coxeter (1969, p. 192) axiomatizes affine geometry (over the reals) as ordered geometry together with an affine form of Desargues’s theorem and an axiom stating that in a plane there is at most one line through a given point not meeting a given line. Affine planes satisfy the following axioms (Cameron 1991, chapter 2): (in which two lines are called parallel if they are equal or disjoint):

• Any two distinct points lie on a unique line.

• Given a point and line there is a unique line which contains the point and is parallel to the line

• There exist three non-collinear points.

As well as affine planes over fields (or division rings), there are also many non-Desarguesian planes satisfying these axioms. (Cameron 1991, chapter 3) gives axioms for higher-dimensional affine spaces.

6.8 Relation to projective spaces

See also: affine space (algebraic geometry) Affine spaces are subspaces of projective spaces: an affine plane can be obtained from any projective plane by

An affine space is a subspace of projective space, which is in turn a quotient of a vector space. 6.9. SEE ALSO 25 removing a line and all the points on it, and conversely any affine plane can be used to construct a projective plane as a closure by adding a line at infinity whose points correspond to equivalence classes of parallel lines. Further, transformations of projective space that preserve affine space (equivalently, that leave the hyperplane at in- finity invariant as a set) yield transformations of affine space. Conversely, any affine linear transformation extends uniquely to a projective linear transformation, so the affine group is a subgroup of the projective group. For instance, Möbius transformations (transformations of the complex projective line, or Riemann sphere) are affine (transforma- tions of the complex plane) if and only if they fix the point at infinity. However, one cannot take the projectivization of an affine space, so projective spaces are not naturally quotients of affine spaces: one can only take the projectivization of a vector space, since the projective space is lines through a given point, and there is no distinguished point in an affine space. If one chooses a base point (as zero), then an affine space becomes a vector space, which one may then projectivize, but this requires a choice.

6.9 See also

• Space (mathematics)

• Affine geometry

• Affine group

• Affine transformation

• Affine variety

• Affine hull

• Heap (mathematics)

• Equipollence (geometry)

• Interval measurement, an affine observable in statistics

• Exotic affine space

• Complex affine space

6.10 Notes

[1] Berger 1987, p. 32

[2] Berger, Marcel (1984), “Affine spaces”, Problems in Geometry, p. 11, ISBN 9780387909714

[3] Berger 1987, p. 33

[4] Snapper, Ernst; Troyer, Robert J. (1989), Metric Affine Geometry, p. 6

[5] Tarrida, Agusti R. (2011), “Affine spaces”, Affine Maps, Euclidean Motions and Quadrics, pp. 1–2, ISBN 9780857297105

[6] Nomizu & Sasaki 1994, p. 7

6.11 References

• Berger, Marcel (1984), “Affine spaces”, Problems in Geometry, Springer-Verlag, ISBN 978-0-387-90971-4

• Berger, Marcel (1987), Geometry I, Berlin: Springer, ISBN 3-540-11658-3

• Cameron, Peter J. (1991), Projective and polar spaces, QMW Maths Notes 13, London: Queen Mary and Westfield College School of Mathematical Sciences, MR 1153019 26 CHAPTER 6. AFFINE SPACE

• Coxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50458-0, MR 123930 • Dolgachev, I.V.; Shirokov, A.P. (2001), “A/a011100”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Snapper, Ernst; Troyer, Robert J. (1989), Metric Affine Geometry (Dover edition, first published in 1989 ed.), Dover Publications, ISBN 0-486-66108-3 • Nomizu, K.; Sasaki, S. (1994), Affine Differential Geometry (New ed.), Cambridge University Press, ISBN 978-0-521-44177-3 • Tarrida, Agusti R. (2011), “Affine spaces”, Affine Maps, Euclidean Motions and Quadrics, Springer, ISBN 978-0-85729-709-9 Chapter 7

Affine transformation

In geometry, an affine transformation, affine map[1] or an affinity (from the Latin, affinis, “connected with”) is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence. If X and Y are affine spaces, then every affine transformation f : X → Y is of the form x 7→ Mx + b , where M is a linear transformation on X and b is a vector in Y . Unlike a purely linear transformation, an affine map need not preserve the zero point in a linear space. Thus, every linear transformation is affine, but not every affine transformation is linear. For many purposes an affine space can be thought of as Euclidean space, though the concept of affine space is far more general (i.e., all Euclidean spaces are affine, but there are affine spaces that are non-Euclidean). In affine coordinates, which include Cartesian coordinates in Euclidean spaces, each output coordinate of an affine map is a linear function (in the sense of calculus) of all input coordinates. Another way to deal with affine transformations systematically is to select a point as the origin; then, any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation.

7.1 Mathematical definition

An affine map[1] f : A → B between two affine spaces is a map on the points that acts linearly on the vectors (that is, the vectors between points of the space). In symbols, f determines a linear transformation φ such that, for any pair of points P,Q ∈ A :

−−−−−−−→ −−→ f(P ) f(Q) = φ(PQ)

or

f(Q) − f(P ) = φ(Q − P )

We can interpret this definition in a few other ways, as follows. If an origin O ∈ A is chosen, and B denotes its image f(O) ∈ B , then this means that for any vector ⃗x :

f :(O + ⃗x) 7→ (B + φ(⃗x)).

If an origin O′ ∈ B is also chosen, this can be decomposed as an affine transformation g : A → B that sends O 7→ O′ , namely

27 28 CHAPTER 7. AFFINE TRANSFORMATION

An image of a fern-like fractal that exhibits affine self-similarity. Each of the leaves of the fern is related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation.

g :(O + ⃗x) 7→ (O′ + φ(⃗x)), 7.2. REPRESENTATION 29

−−→ followed by the translation by a vector ⃗b = O′B . The conclusion is that, intuitively, f consists of a translation and a linear map.

7.1.1 Alternative definition

Given two affine spaces A and B , over the same field, a function f : A → B is an affine map if and only if for every family {(ai, λi)}i∈I of weighted points in A such that

∑ λi = 1, i∈I we have[2]

( ) ∑ ∑ f λiai = λif(ai) . i∈I i∈I In other words, f preserves barycenters.

7.2 Representation

As shown above, an affine map is the composition of two functions: a translation and a linear map. Ordinary vector algebra uses matrix multiplication to represent linear maps, and vector addition to represent translations. Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by a matrix A and the translation as the addition of a vector ⃗b , an affine map f acting on a vector ⃗x can be represented as

⃗y = f(⃗x) = A⃗x +⃗b.

7.2.1 Augmented matrix

Using an augmented matrix and an augmented vector, it is possible to represent both the translation and the linear map using a single matrix multiplication. The technique requires that all vectors are augmented with a “1” at the end, and all matrices are augmented with an extra row of zeros at the bottom, an extra column—the translation vector—to the right, and a “1” in the lower right corner. If A is a matrix,

[ ] [ ][ ] ⃗y A ⃗b ⃗x = 1 0 ... 0 1 1

is equivalent to the following

⃗y = A⃗x +⃗b.

The above-mentioned augmented matrix is called affine transformation matrix, or projective transformation matrix (as it can also be used to perform projective transformations). This representation exhibits the set of all invertible affine transformations as the semidirect product of Kn and GL(n, k). This is a group under the operation of composition of functions, called the affine group. Ordinary matrix-vector multiplication always maps the origin to the origin, and could therefore never represent a translation, in which the origin must necessarily be mapped to some other point. By appending the additional coor- dinate “1” to every vector, one essentially considers the space to be mapped as a subset of a space with an additional dimension. In that space, the original space occupies the subset in which the additional coordinate is 1. Thus the 30 CHAPTER 7. AFFINE TRANSFORMATION

Affine transformations on the 2D plane can be performed in three dimensions. Translation is done by shearing along over the z axis, and rotation is performed around the z axis. origin of the original space can be found at (0,0, ... 0, 1). A translation within the original space by means of a linear transformation of the higher-dimensional space is then possible (specifically, a shear transformation). The coordinates in the higher-dimensional space are an example of homogeneous coordinates. If the original space is Euclidean, the higher dimensional space is a real projective space. The advantage of using homogeneous coordinates is that one can combine any number of affine transformations into one by multiplying the respective matrices. This property is used extensively in computer graphics, computer vision and robotics.

Example augmented matrix

If the vectors ⃗x1, . . . , ⃗xn+1 are a basis of the domain’s projective vector space and if ⃗y1, . . . , ⃗yn+1 are the corre- sponding vectors in the codomain vector space then the augmented matrix M that achieves this affine transformation

[ ] [ ] ⃗y ⃗x = M 1 1 is 7.3. PROPERTIES 31

[ ][ ]−1 ⃗y . . . ⃗y ⃗x . . . ⃗x M = 1 n+1 1 n+1 1 ... 1 1 ... 1

This formulation works irrespective of whether any of the domain, codomain and image vector spaces have the same number of dimensions. For example, the affine transformation of a vector plane is uniquely determined from the knowledge of where the three vertices of a non-degenerate triangle are mapped to.

7.3 Properties

An affine transformation preserves:

1. The collinearity relation between points; i.e., points which lie on the same line (called collinear points) continue to be collinear after the transformation. −−→ −−→ 2. Ratios of vectors−−−−−−−→ along a line; i.e.,−−−−−−−→ for distinct collinear points p1, p2, p3, the ratio of p1p2 and p2p3 is the same as that of f(p1)f(p2) and f(p2)f(p3) . 3. More generally barycenters of weighted collections of points.

An affine transformation is invertible if and only if A is invertible. In the matrix representation, the inverse is:

[ ] A−1 −A−1⃗b 0 ... 0 1

The invertible affine transformations (of an affine space onto itself) form the affine group, which has the general linear group of degree n as subgroup and is itself a subgroup of the general linear group of degree n + 1. The similarity transformations form the subgroup where A is a scalar times an orthogonal matrix. For example, if the affine transformation acts on the plane and if the determinant of A is 1 or −1 then the transformation is an equi- areal mapping. Such transformations form a subgroup called the equi-affine group[3] A transformation that is both equi-affine and a similarity is an isometry of the plane taken with Euclidean distance. Each of these groups has a subgroup of transformations which preserve orientation: those where the determinant of A is positive. In the last case this is in 3D the group of rigid body motions (proper rotations and pure translations). If there is a fixed point, we can take that as the origin, and the affine transformation reduces to a linear transformation. This may make it easier to classify and understand the transformation. For example, describing a transformation as a rotation by a certain angle with respect to a certain axis may give a clearer idea of the overall behavior of the transformation than describing it as a combination of a translation and a rotation. However, this depends on application and context.

7.4 Affine transformation of the plane

Affine transformations in two real dimensions include:

• pure translations, • scaling in a given direction, with respect to a line in another direction (not necessarily perpendicular), combined with translation that is not purely in the direction of scaling; taking “scaling” in a generalized sense it includes the cases that the scale factor is zero (projection) or negative; the latter includes reflection, and combined with translation it includes glide reflection, • rotation combined with a homothety and a translation, • shear mapping combined with a homothety and a translation, or 32 CHAPTER 7. AFFINE TRANSFORMATION

A1 A2 Z B2 B1 C2

C1

A central dilation. The triangles A1B1Z, A1C1Z, and B1C1Z get mapped to A2B2Z, A2C2Z, and B2C2Z, respectively.

• squeeze mapping combined with a homothety and a translation.

To visualise the general affine transformation of the Euclidean plane, take labelled parallelograms ABCD and A′B′C′D′. Whatever the choices of points, there is an affine transformation T of the plane taking A to A′, and each vertex sim- ilarly. Supposing we exclude the degenerate case where ABCD has zero area, there is a unique such affine transfor- mation T. Drawing out a whole grid of parallelograms based on ABCD, the image T(P) of any point P is determined by noting that T(A) = A′, T applied to the line segment AB is A′B′, T applied to the line segment AC is A′C′, and T respects scalar multiples of vectors based at A. [If A, E, F are collinear then the ratio length(AF)/length(AE) is equal to length(A′F′)/length(A′E′).] Geometrically T transforms the grid based on ABCD to that based in A′B′C′D′. Affine transformations don't respect lengths or angles; they multiply area by a constant factor

area of A′B′C′D′ / area of ABCD.

A given T may either be direct (respect orientation), or indirect (reverse orientation), and this may be determined by its effect on signed areas (as defined, for example, by the cross product of vectors).

7.5 Examples of affine transformations

7.5.1 Affine transformations over the real numbers

Functions f : R → R, f(x) = mx + c with m and c constant, are commonplace affine transformations.

7.5.2 Affine transformation over a finite field

The following equation expresses an affine transformation in GF(28):

{ a′ } = M{ a } ⊕ { v },

For instance, the affine transformation of the element {a} = y7 + y6 + y3 + y = {11001010} in big-endian binary notation = {CA} in big-endian hexadecimal notation, is calculated as follows: 7.6. SEE ALSO 33

′ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ a0 = a0 a4 a5 a6 a7 1 = 0 0 0 1 1 1 = 1

′ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ a1 = a0 a1 a5 a6 a7 1 = 0 1 0 1 1 1 = 0 ′ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ a2 = a0 a1 a2 a6 a7 0 = 0 1 0 1 1 0 = 1 ′ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ a3 = a0 a1 a2 a3 a7 0 = 0 1 0 1 1 0 = 1 ′ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ a4 = a0 a1 a2 a3 a4 0 = 0 1 0 1 0 0 = 0 ′ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ a5 = a1 a2 a3 a4 a5 1 = 1 0 1 0 0 1 = 1 ′ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ a6 = a2 a3 a4 a5 a6 1 = 0 1 0 0 1 1 = 1 ′ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ a7 = a3 a4 a5 a6 a7 0 = 1 0 0 1 1 0 = 1. Thus, {a′} = y7 + y6 + y5 + y3 + y2 + 1 = {11101101} = {ED}.

7.5.3 Affine transformation in plane geometry

In ℝ2, the transformation shown at left is accomplished using the map given by:

[ ] [ ][ ] [ ] x 0 1 x −100 7→ + y 2 1 y −100

Transforming the three corner points of the original triangle (in red) gives three new points which form the new triangle (in blue). This transformation skews and translates the original triangle. In fact, all triangles are related to one another by affine transformations. This is also true for all parallelograms, but not for all quadrilaterals.

7.6 See also

• The transformation matrix for an affine transformation

• Affine geometry

• 3D projection

• Homography

• Flat (geometry)

• Bent function

7.7 Notes

[1] Berger, Marcel (1987), p. 38.

[2] Schneider, Philip K. & Eberly, David H. (2003). Geometric Tools for Computer Graphics. Morgan Kaufmann. p. 98. ISBN 978-1-55860-594-7.

[3] Oswald Veblen (1918) Projective Geometry, volume 2, pp. 105–7. 34 CHAPTER 7. AFFINE TRANSFORMATION

A simple affine transformation on the real plane

7.8 References

• Berger, Marcel (1987), Geometry I, Berlin: Springer, ISBN 3-540-11658-3

• Nomizu, Katsumi; Sasaki, S. (1994), Affine Differential Geometry (New ed.), Cambridge University Press, ISBN 978-0-521-44177-3

• Sharpe, R. W. (1997). Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program. New York: Springer. ISBN 0-387-94732-9. 7.9. EXTERNAL LINKS 35

No change Translate Scale about origin 1 0 0 1 0 X W 0 0 0 1 0 0 1 Y 0 H 0 0 0 1 0 0 1 0 0 1 y y y (0,1) (0,H) 1 1 1 (0,0) (1,0) (X,Y) (W,0) 0 1x 0 1x 0 1x

Rotate about origin Shear in x direction Shear in y direction cos θ sin θ 0 1 A 0 1 0 0 -sin θ cos θ 0 0 1 0 B 1 0 0 0 1 0 0 1 0 0 1 y y y (sin θ, cos θ) (A,1) (0,1) 1 1 1 θ (1,0) (1,B) 0 1x 0 1x 0 1x (cos θ, -sin θ)

Reflect about origin Reflect about x-axis Reflect about y-axis -1 0 0 1 0 0 -1 0 0 0 -1 0 0 -1 0 0 1 0 0 0 1 0 0 1 0 0 1 y y y (0,1) 1 1 1 (-1,0) (1,0) 0 1x 0 1x (-1,0) 0 1x (0,-1) (0,-1)

Effect of applying various 2D affine transformation matrices on a unit square. Note that the reflection matrices are special cases of the scaling matrix.

7.9 External links

• Hazewinkel, Michiel, ed. (2001), “Affine transformation”, Encyclopedia of Mathematics, Springer, ISBN 978- 1-55608-010-4

• Geometric Operations: Affine Transform, R. Fisher, S. Perkins, A. Walker and E. Wolfart. 36 CHAPTER 7. AFFINE TRANSFORMATION

• Weisstein, Eric W., “Affine Transformation”, MathWorld.

• Affine Transform by Bernard Vuilleumier, Wolfram Demonstrations Project. • Affine Transformation with MATLAB Chapter 8

Augmented matrix

In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices. Given the matrices A and B, where     1 3 2 4 A = 2 0 1,B = 3, 5 2 2 1 the augmented matrix (A|B) is written as   1 3 2 4 (A|B) =  2 0 1 3  . 5 2 2 1 This is useful when solving systems of linear equations. For a given number of unknowns, the number of solutions to a system of linear equations depends only on the rank of the matrix representing the system and the rank of the corresponding augmented matrix. Specifically, according to the Rouché–Capelli theorem, any system of linear equations is inconsistent (has no solutions) if the rank of the augmented matrix is greater than the rank of the coefficient matrix; if, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank; hence in such a case there are an infinitude of solutions. An augmented matrix may also be used to find the inverse of a matrix by combining it with the identity matrix.

8.1 Examples

8.1.1 Matrix inverse

Let C be the square 2×2 matrix

[ ] 1 3 C = . −5 0 To find the inverse of C we create (C|I) where I is the 2×2 identity matrix. We then reduce the part of (C|I) corre- sponding to C to the identity matrix using only elementary row operations on (C|I).

[ ] 1 3 1 0 (C|I) = −5 0 0 1 [ ] − 1 −1 1 0 0 5 (I|C ) = 1 1 0 1 3 15

37 38 CHAPTER 8. AUGMENTED MATRIX the right part of which is the inverse of the original matrix.

8.1.2 Existence and number of solutions

Consider the system of equations

x + y + 2z = 3 x + y + z = 1

2x + 2y + 2z = 2.

The coefficient matrix is

  1 1 2 A = 1 1 1, 2 2 2 and the augmented matrix is

  1 1 2 3 (A|B) =  1 1 1 1  . 2 2 2 2

Since both of these have the same rank, namely 2, there exists at least one solution; and since their rank is less than the number of unknowns, the latter being 3, there are an infinite number of solutions. In contrast, consider the system

x + y + 2z = 3 x + y + z = 1 2x + 2y + 2z = 5.

The coefficient matrix is

  1 1 2 A = 1 1 1, 2 2 2 and the augmented matrix is

  1 1 2 3 (A|B) =  1 1 1 1  . 2 2 2 5

In this example the coefficient matrix has rank 2 while the augmented matrix has rank 3; so this system of equations has no solution. Indeed, an increase in the number of linearly independent rows has made the system of equations inconsistent.

8.1.3 Solution of a linear system

As used in linear algebra, an augmented matrix is used to represent the coefficients and the solution vector of each equation set. For the set of equations 8.2. REFERENCES 39

x + 2y + 3z = 0 3x + 4y + 7z = 2 6x + 5y + 9z = 11 the coefficients and constant terms give the matrices

    1 2 3 0 A = 3 4 7,B =  2 , 6 5 9 11 and hence give the augmented matrix

  1 2 3 0 (A|B) =  3 4 7 2  6 5 9 11

Note that the rank of the coefficient matrix, which is 3, equals the rank of the augmented matrix, so at least one solution exists; and since this rank equals the number of unknowns, there is exactly one solution. To obtain the solution, row operations can be performed on the augmented matrix to obtain the identity matrix on the left side, yielding

  1 0 0 4  0 1 0 1  , 0 0 1 −2 so the solution of the system is (x, y, z) = (4, 1, −2).

8.2 References

• Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Dover Publications, 1992, ISBN 0-486-67102-X. Page 31. Chapter 9

Barycenter

Barycentric view of the Pluto–Charon system as seen by New Horizons

The barycenter (or barycentre), (from the Greek βαρύ-ς heavy + κέντρ-ον centre[1]) is the center of mass of two or more bodies that are orbiting each other, or the point around which they both orbit. It is an important concept in fields such as astronomy and astrophysics. The distance from a body’s center of mass to the barycenter can be

40 9.1. TWO-BODY PROBLEM 41

calculated as a simple two-body problem. In cases where one of the two objects is considerably more massive than the other (and relatively close), the barycenter will typically be located within the more massive object. Rather than appearing to orbit a common center of mass with the smaller body, the larger will simply be seen to “wobble” slightly. This is the case for the Earth–Moon system, where the barycenter is located on average 4,671 km from the Earth’s center, well within the planet’s radius of 6,378 km. When the two bodies are of similar masses, the barycenter will generally be located between them and both bodies will follow an orbit around it. This is the case for Pluto and Charon, as well as for many binary asteroids and binary stars. It is also the case for Jupiter and the Sun, despite the 1,000-fold difference in mass, due to the relatively large distance between them. In astronomy, barycentric coordinates are non-rotating coordinates with the origin at the center of mass of two or more bodies. The International Celestial Reference System is a barycentric one, based on the barycenter of the Solar System. In geometry, the term “barycenter” is synonymous with centroid, the geometric center of a two-dimensional shape.

9.1 Two-body problem

Main article: Two-body problem The barycenter is one of the foci of the elliptical orbit of each body. This is an important concept in the fields of astronomy, astrophysics. If a is the distance between the centers of the two bodies (the semi-major axis of the system), r1 is the semi-major axis of the primary’s orbit around the barycenter, and r2 = a − r1 is the semi-major axis of the secondary’s orbit. When the barycenter is located within the more massive body, that body will appear to “wobble” rather than to follow a discernible orbit. In a simple two-body case, r1, the distance from the center of the primary to the barycenter is given by:

m2 a r1 = a · = m1 + m2 1 + m1/m2 where :

r1 is the distance from body 1 to the barycenter a is the distance between the centers of the two bodies

m1 and m2 are the masses of the two bodies.

9.1.1 Primary–secondary examples

The following table sets out some examples from the Solar System. Figures are given rounded to three significant figures. The last two columns show R1, the radius of the first (more massive) body, and r1 / R1, the ratio of the distance to the barycenter and that radius: a value less than one shows that the barycenter lies inside the first body. The term primary–secondary is used to distinguish between the different degrees of relationship of the involved participants.

9.1.2 Inside or outside the Sun?

If m1 ≫ m2 — which is true for the Sun and any planet — then the ratio r1/R1 approximates to:

a · m2 R1 m1 Hence, the barycenter of the Sun–planet system will lie outside the Sun only if:

a m · planet > 1 ⇒ a · m > R⊙ · m⊙ ≈ 2.3 × 1011 m km ≈ 1530 m AU R⊙ m⊙ planet Earth Earth 42 CHAPTER 9. BARYCENTER

1984

Center of Mass

1994 1959

1973

1988 1948 1950 1981

1951 1991 1964 1990 Nucleus of Sun

1952 1968

1955

1978 1945 Limb of Sun

Motion of the Solar System's barycenter relative to the Sun

That is, where the planet is heavy and far from the Sun. If Jupiter had Mercury's orbit (57,900,000 km, 0.387 AU), the Sun–Jupiter barycenter would be approximately 55,000 km from the center of the Sun (r1/R1 ~ 0.08). But even if the Earth had Eris’ orbit (68 AU), the Sun–Earth barycenter would still be within the Sun (just over 30,000 km from the center). To calculate the actual motion of the Sun, you would need to sum all the influences from all the planets, comets, asteroids, etc. of the Solar System (see n-body problem). If all the planets were aligned on the same side of the Sun, the combined center of mass would lie about 500,000 km above the Sun’s surface.

The calculations above are based on the mean distance between the bodies and yield the mean value r1. But all celestial orbits are elliptical, and the distance between the bodies varies between the apses, depending on the eccentricity, e. Hence, the position of the barycenter varies too, and it is possible in some systems for the barycenter to be sometimes inside and sometimes outside the more massive body. This occurs where:

1 r 1 > 1 > 1 − e R1 1 + e

Note that the Sun–Jupiter system, with eJᵤᵢₑᵣ = 0.0484, just fails to qualify: 1.05 ≯ 1.07 > 0.954. 9.2. GALLERY 43

9.2 Gallery

Images are representative (made by hand), not simulated.

• Two bodies with the same mass orbiting a common barycenter (similar to the 90 Antiope system)

• Two bodies with a difference in mass orbiting a common barycenter external to both bodies, as in the Pluto– Charon system

• Two bodies with a major difference in mass orbiting a common barycenter internal to one body (similar to the Earth–Moon system)

• Two bodies with an extreme difference in mass orbiting a common barycenter internal to one body (similar to the Sun–Earth system)

• Two bodies with the same mass orbiting a common barycenter, external to both bodies, with eccentric elliptic orbits (a common situation for binary stars)

• Scale model of the Pluto system: Pluto and its five moons, including the location of the system’s barycenter. Sizes, distances and apparent magnitude of the bodies are to scale.

• Sideview of a star orbiting the barycenter of a planetary system. The radial-velocity method makes use of the star’s wobble to detect extrasolar planets

9.3 Relativistic corrections

In classical mechanics, this definition simplifies calculations and introduces no known problems. In general relativity, problems arise because, while it is possible, within reasonable approximations, to define the barycenter, the associated coordinate system does not fully reflect the inequality of clock rates at different locations. Brumberg explains how to set up barycentric coordinates in general relativity.[3] The coordinate systems involve a world-time, i.e. a global time coordinate that could be set up by telemetry. Individual clocks of similar construction will not agree with this standard, because they are subject to differing gravitational potentials or move at various velocities, so the world-time must be slaved to some ideal clock that is assumed to be very far from the whole self-gravitating system. This time standard is called Barycentric Coordinate Time, “TCB”.

9.4 Selected barycentric orbital elements

Barycentric osculating orbital elements for some objects in the Solar System:[4] For objects at such high eccentricity, the Sun’s barycentric coordinates are more stable than heliocentric coordinates.[5]

9.5 See also

• Buoyancy • Centers of gravity in non-uniform fields • Center of mass • Center of mass (relativistic) 44 CHAPTER 9. BARYCENTER

• Center of percussion

• Centroid • Expected value

• Mass point geometry • Metacentric height

• Roll center • Weight distribution

9.6 References

[1] Oxford English Dictionary, Second Edition.

[2] “What’s a Barycenter?". Space Place @ NASA. 2005-09-08. Archived from the original on 23 December 2010. Retrieved 2011-01-20.

[3] Essential Relativistic Celestial Mechanics by Victor A. Brumberg (Adam Hilger, London, 1991) ISBN 0-7503-0062-0.

[4] Horizons output (2011-01-30). “Barycentric Osculating Orbital Elements for 2007 TG422”. Retrieved 2011-01-31. (Se- lect Ephemeris Type:Elements and Center:@0)

[5] Kaib, Nathan A.; Becker, Andrew C.; Jones, R. Lynne; Puckett, Andrew W.; Bizyaev, Dmitry; Dilday, Benjamin; Frieman, Joshua A.; Oravetz, Daniel J.; Pan, Kaike; Quinn, Thomas; Schneider, Donald P.; Watters, Shannon (2009). “2006 SQ372: A Likely Long-Period Comet from the Inner Oort Cloud”. The Astrophysical Journal 695 (1): 268–275. arXiv:0901.1690. Bibcode:2009ApJ...695..268K. doi:10.1088/0004-637X/695/1/268. Chapter 10

Bent function

The four 2-ary Boolean functions with Hamming weight 1 are bent, i.e. their nonlinearity is 1 (which is what this diagram shows). The following formula shows that a 2-ary function is bent when its nonlinearity is 1: 2−1 2 −1 2 − 2 2 = 2 − 1 = 1

In the mathematical field of combinatorics, a bent function is a special type of Boolean function. This means it takes several inputs and gives one output, each of which has two possible values (such as 0 and 1, or true and false). The name is figurative. Bent functions are so called because they are as different as possible from all linear and affine functions, the simplest or “straight” functions. This makes the bent functions naturally hard to approximate. Bent functions were defined and named in the 1960s by Oscar Rothaus in research not published until 1976.[1] They have been extensively studied for their applications in cryptography, but have also been applied to spread spectrum, coding theory, and combinatorial design. The definition can be extended in several ways, leading to different classes of generalized bent functions that share many of the useful properties of the original.

10.1 Walsh transform

Bent functions are defined in terms of the Walsh transform. The Walsh transform of a Boolean function ƒ: Zn ˆ Zn → Z 2 → Z2 is the function f : 2 given by

45 46 CHAPTER 10. BENT FUNCTION

∑ fˆ(a) = (−1)f(x)+a·x ∈Zn x 2

where a · x = a1x1 + a2x2 + … + anxn (mod 2) is the dot product in Zn [2] 2. Alternatively, let S0(a) = { x ∈ Zn 2 : ƒ(x) = a · x } and S1(a) = { x ∈ Zn n 2 : ƒ(x) ≠ a · x }. Then |S0(a)| + |S1(a)| = 2 and hence

ˆ n f(a) = |S0(a)| − |S1(a)| = 2|S0(a)| − 2 .

For any Boolean function ƒ and a ∈ Zn 2 the transform lies in the range

−2n ≤ fˆ(a) ≤ 2n.

Moreover, the linear function ƒ0(x) = a · x and the affine function ƒ1(x) = a · x + 1 correspond to the two extreme cases, since

ˆ n ˆ n f0(a) = 2 , f1(a) = −2 .

Thus, for each a ∈ Zn ˆ 2 the value of f(a) characterizes where the function ƒ(x) lies in the range from ƒ0(x) to ƒ1(x).

10.2 Definition and properties

Rothaus defined a bent function as a Boolean function ƒ: Zn 2 → Z2 whose Walsh transform has constant absolute value. Bent functions are in a sense equidistant from all the affine functions, so they are equally hard to approximate with any affine function.

The simplest examples of bent functions, written in algebraic normal form, are F(x1,x2) = x1x2 and G(x1,x2,x3,x4) = x1x2 + x3x4. This pattern continues: x1x2 + x3x4 + ... + xn ₋ ₁xn is a bent function Zn [3] 2 → Z2 for every even n, but there is a wide variety of different types of bent functions as n increases. The sequence of values (−1)ƒ(x), with x ∈ Zn 2 taken in lexicographical order, is called a bent sequence; bent functions and bent sequences have equivalent prop- erties. In this ±1 form, the Walsh transform is easily computed as

fˆ(a) = W (2n)(−1)f(a),

where W(2n) is the natural-ordered Walsh matrix and the sequence is treated as a column vector.[4] Rothaus proved that bent functions exist only for even n, and that for a bent function ƒ, |fˆ(a)| = 2n/2 for all a ∈ Zn 2.[2] In fact, fˆ(a) = 2n/2(−1)g(a) , where g is also bent. In this case, gˆ(a) = 2n/2(−1)f(a) , so ƒ and g are considered dual functions.[4] Every bent function has a Hamming weight (number of times it takes the value 1) of 2n − 1 ± 2n/2 − 1, and in fact agrees with any affine function at one of those two numbers of points. So the nonlinearity of ƒ (minimum number of times it equals any affine function) is 2n − 1 − 2n/2 − 1, the maximum possible. Conversely, any Boolean function with nonlinearity 2n − 1 − 2n/2 − 1 is bent.[2] The degree of ƒ in algebraic normal form (called the nonlinear order of ƒ) is at most n/2 (for n > 2).[3] Although bent functions are vanishingly rare among Boolean functions of many variables, they come in many different kinds. There has been detailed research into special classes of bent functions, such as the homogeneous ones[5] or those arising from a monomial over a finite field,[6] but so far the bent functions have defied all attempts at a complete enumeration or classification. 10.3. APPLICATIONS 47

10.3 Applications

As early as 1982 it was discovered that maximum length sequences based on bent functions have cross-correlation and autocorrelation properties rivalling those of the Gold codes and Kasami codes for use in CDMA.[7] These sequences have several applications in spread spectrum techniques. The properties of bent functions are naturally of interest in modern digital cryptography, which seeks to obscure relationships between input and output. By 1988 Forré recognized that the Walsh transform of a function can be used to show that it satisfies the Strict Avalanche Criterion (SAC) and higher-order generalizations, and recommended this tool to select candidates for good S-boxes achieving near-perfect diffusion.[8] Indeed, the functions satisfying the SAC to the highest possible order are always bent.[9] Furthermore, the bent functions are as far as possible from having what are called linear structures, nonzero vectors a such that ƒ(x+a) + ƒ(x) is a constant. In the language of differential cryptanalysis (introduced after this property was discovered) the derivative of a bent function ƒ at every nonzero point a (that is, ƒa(x) = ƒ(x+a) + ƒ(x)) is a balanced Boolean function, taking on each value exactly half of the time. This property is called perfect nonlinearity.[3] Given such good diffusion properties, apparently perfect resistance to differential cryptanalysis, and resistance by definition to linear cryptanalysis, bent functions might at first seem the ideal choice for secure cryptographic functions such as S-boxes. Their fatal flaw is that they fail to be balanced. In particular, an invertible S-box cannot be constructed directly from bent functions, and a stream cipher using a bent combining function is vulnerable to a correlation attack. Instead, one might start with a bent function and randomly complement appropriate values until the result is balanced. The modified function still has high nonlinearity, and as such functions are very rare the process should be much faster than a brute-force search.[3] But functions produced in this way may lose other desirable properties, even failing to satisfy the SAC—so careful testing is necessary.[9] A number of cryptographers have worked on techniques for generating balanced functions that preserve as many of the good cryptographic qualities of bent functions as possible.[10][11][12] Some of this theoretical research has been incorporated into real cryptographic algorithms. The CAST design pro- cedure, used by Carlisle Adams and Stafford Tavares to construct the S-boxes for the block ciphers CAST-128 and CAST-256, makes use of bent functions.[12] The cryptographic hash function HAVAL uses Boolean functions built from representatives of all four of the equivalence classes of bent functions on six variables.[13] The stream cipher Grain uses an NLFSR whose nonlinear feedback polynomial is, by design, the sum of a bent function and a linear function.[14]

10.4 Generalizations

Zn → Z The most common class of generalized bent functions is the mod m type, f : m m such that

∑ 2πi (f(x)−a·x) fˆ(a) = e m ∈Zn x m n/2 Zn → Z has constant absolute value m . Perfect nonlinear functions f : m m , those such that for all nonzero a, ƒ(x+a) − ƒ(a) takes on each value mn − 1 times, are generalized bent. If m is prime, the converse is true. In most cases only prime m are considered. For odd prime m, there are generalized bent functions for every positive n, even and odd. They have many of the same good cryptographic properties as the binary bent functions.[15] Zn → Z Semi-bent functions are an odd-order counterpart to bent functions. A semi-bent function is f : m m with n odd, such that |fˆ| takes only the values 0 and m(n+1)/2. They also have good cryptographic characteristics, and some of them are balanced, taking on all possible values equally often.[16] The partially bent functions form a large class defined by a condition on the Walsh transform and autocorrela- tion functions. All affine and bent functions are partially bent. This is in turn a proper subclass of the plateaued functions.[17] The idea behind the hyper-bent functions is to maximize the minimum distance to all Boolean functions coming from bijective monomials on the finite field GF(2n), not just the affine functions. For these functions this distance is constant, which may make them resistant to an interpolation attack. Other related names have been given to cryptographically important classes of functions Zn 2 → Zn 48 CHAPTER 10. BENT FUNCTION

2, such as almost bent functions and crooked functions. While not bent functions themselves (these are not even Boolean functions), they are closely related to the bent functions and have good nonlinearity properties.

10.5 References

[1] O. S. Rothaus (May 1976). “On “Bent” Functions”. Journal of Combinatorial Theory, Series A 20 (3): 300–305. doi:10.1016/0097- 3165(76)90024-8. ISSN 0097-3165. Retrieved 16 December 2013. [2] C. Qu; J. Seberry; T. Xia (29 December 2001). “Boolean Functions in Cryptography”. Retrieved 14 September 2009. [3] W. Meier; O. Staffelbach (April 1989). Nonlinearity Criteria for Cryptographic Functions. Eurocrypt '89. pp. 549–562. [4] C. Carlet; L.E. Danielsen; M.G. Parker; P. Solé (19 May 2008). Self Dual Bent Functions (PDF). Fourth International Workshop on Boolean Functions: Cryptography and Applications (BFCA '08). Retrieved 21 September 2009. [5] T. Xia; J. Seberry; J. Pieprzyk; C. Charnes (June 2004). “Homogeneous bent functions of degree n in 2n variables do not exist for n > 3”. Discrete Applied Mathematics 142 (1–3): 127–132. doi:10.1016/j.dam.2004.02.006. ISSN 0166-218X. Retrieved 21 September 2009. [6] A. Canteaut; P. Charpin; G. Kyureghyan (January 2008). “A new class of monomial bent functions” (PDF). Finite Fields and Their Applications 14 (1): 221–241. doi:10.1016/j.ffa.2007.02.004. ISSN 1071-5797. Retrieved 21 September 2009. [7] J. Olsen; R. Scholtz; L. Welch (November 1982). “Bent-Function Sequences”. IEEE Transactions on Information Theory. IT-28 (6): 858–864. ISSN 0018-9448. Retrieved 24 September 2009. [8] R. Forré (August 1988). The Strict Avalanche Criterion: Spectral Properties of Boolean Functions and an Extended Defini- tion. CRYPTO '88. pp. 450–468. [9] C. Adams; S. Tavares (January 1990). “The Use of Bent Sequences to Achieve Higher-Order Strict Avalanche Criterion in S-Box Design”. Technical Report TR 90-013. Queen’s University. CiteSeerX: 10 .1 .1 .41 .8374. [10] K. Nyberg (April 1991). Perfect nonlinear S-boxes. Eurocrypt '91. pp. 378–386. [11] J. Seberry; X. Zhang (December 1992). Highly Nonlinear 0-1 Balanced Boolean Functions Satisfying Strict Avalanche Criterion. AUSCRYPT '92. pp. 143–155. CiteSeerX: 10 .1 .1 .57 .4992. [12] C. Adams (November 1997). “Constructing Symmetric Ciphers Using the CAST Design Procedure”. Designs, Codes, and Cryptography 12 (3): 283–316. doi:10.1023/A:1008229029587. ISSN 0925-1022. Retrieved 20 September 2009. [13] Y. Zheng; J. Pieprzyk; J. Seberry (December 1992). HAVAL—a one-way hashing algorithm with variable length of output. AUSCRYPT '92. pp. 83–104. Retrieved 20 June 2015. [14] M. Hell; T. Johansson; A. Maximov; W. Meier. “A Stream Cipher Proposal: Grain-128” (PDF). Retrieved 24 September 2009. [15] K. Nyberg (May 1990). Constructions of bent functions and difference sets. Eurocrypt '90. pp. 151–160. [16] K. Khoo; G. Gong; D. Stinson (February 2006). “A new characterization of semi-bent and bent functions on finite fields” (PostScript). Designs, Codes, and Cryptography 38 (2): 279–295. doi:10.1007/s10623-005-6345-x. ISSN 0925-1022. Retrieved 24 September 2009. [17] Y. Zheng; X. Zhang (November 1999). Plateaued Functions. Second International Conference on Information and Com- munication Security (ICICS '99). pp. 284–300. Retrieved 24 September 2009.

10.6 Further reading

• C. Carlet (May 1993). Two New Classes of Bent Functions. Eurocrypt '93. pp. 77–101. • J. Seberry; X. Zhang (March 1994). “Constructions of Bent Functions from Two Known Bent Functions”. Australasian Journal of Combinatorics 9: 21–35. ISSN 1034-4942. CiteSeerX: 10 .1 .1 .55 .531. • T. Neumann (May 2006). “Bent Functions”. CiteSeerX: 10 .1 .1 .85 .8731. • Colbourn, Charles J.; Dinitz, Jeffrey H. (2006). Handbook of Combinatorial Designs (2nd ed.). CRC Press. pp. 337–339. ISBN 978-1-58488-506-1. • Cusick, T.W.; Stanica, P. (2009). Cryptographic Boolean Functions and Applications. Academic Press. ISBN 9780123748904. 10.6. FURTHER READING 49

The Boolean function x1x2 + x3x4 is bent, i.e. its nonlinearity is 6 (which is what this diagram shows). The following formula shows that a 4-ary function is bent when its nonlinearity is 6: 4−1 4 −1 2 − 2 2 = 8 − 2 = 6 + stands for the exclusive or (compare algebraic normal form) Chapter 11

Cartesian coordinate system

y (2,3) 3

2 (−3,1) 1 (0,0) x −3−2−1 1 2 3 −1

−2

(−1.5,−2.5) −3

Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines,

50 51

measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular pro- jection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. y

3

2 2 2 x + y = 4 1

-3 -2 -1 1 2 3 x -1

-2

-3

Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (x − a)2 + (y − b)2 = r2 where a and b are the coordinates of the center (a, b) and r is the radius.

The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revo- lutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 in a plane may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4. Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates 52 CHAPTER 11. CARTESIAN COORDINATE SYSTEM are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.

11.1 History

The adjective Cartesian refers to the French mathematician and philosopher René Descartes (who used the name Cartesius in Latin). The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions and did not publish the discovery.[1] Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes’ La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes’ work.[2] The development of the Cartesian coordinate system would play a fundamental role in the development of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz.[3] Nicole Oresme, a French cleric and friend of the Dauphin (later to become King Charles V) of the 14th Century, used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat. Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space.

11.2 Description

11.2.1 One dimension

Main article: Number line

Choosing a Cartesian coordinate system for a one-dimensional space—that is, for a straight line—involves choosing a point O of the line (the origin), a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative; we then say that the line “is oriented” (or “points”) from the negative half towards the positive half. Then each point P of the line can be specified by its distance from O, taken with a + or − sign depending on which half-line contains P. A line with a chosen Cartesian system is called a number line. Every real number has a unique location on the line. Conversely, every point on the line can be interpreted as a number in an ordered continuum such as the real numbers.

11.2.2 Two dimensions

Further information: Two-dimensional space

The Cartesian coordinate system in two dimensions (also called a rectangular coordinate system) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis. (Early systems allowed “oblique” axes, that is, axes that did not meet at right angles.) The lines are commonly referred to as the x- and y-axes where the x-axis is taken to be horizontal and the y-axis is taken to be vertical. The point where the axes meet is taken as the origin for both, thus turning each axis into a number line. For a given point P, a line is drawn through P perpendicular to the x-axis to meet it at X and second line is drawn through P perpendicular to the y-axis to meet it at Y. The coordinates of P are then X and Y interpreted as numbers x and y on the corresponding number lines. The coordinates are written as an ordered pair (x, y). The point where the axes meet is the common origin of the two number lines and is simply called the origin. It is often labeled O and if so then the axes are called Ox and Oy. A plane with x- and y-axes defined is often referred to as the Cartesian plane or xy-plane. The value of x is called the x-coordinate or abscissa and the value of y is called the y-coordinate or ordinate. 11.2. DESCRIPTION 53

The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values. In the Cartesian plane, reference is sometimes made to a unit circle or a unit hyperbola.

11.2.3 Three dimensions

Further information: Three-dimensional space Choosing a Cartesian coordinate system for a three-dimensional space means choosing an ordered triplet of lines

Z

(x,y,z)

O z x y Y

X

A three dimensional Cartesian coordinate system, with origin O and axis lines X, Y and Z, oriented as shown by the arrows. The tick marks on the axes are one length unit apart. The black dot shows the point with coordinates x = 2, y = 3, and z = 4, or (2,3,4).

(axes) that are pair-wise perpendicular, have a single unit of length for all three axes and have an orientation for each axis. As in the two-dimensional case, each axis becomes a number line. The coordinates of a point P are obtained by drawing a line through P perpendicular to each coordinate axis, and reading the points where these lines meet the axes as three numbers of these number lines. Alternatively, the coordinates of a point P can also be taken as the (signed) distances from P to the three planes defined by the three axes. If the axes are named x, y, and z, then the x-coordinate is the distance from the plane defined by the y and z axes. The distance is to be taken with the + or − sign, depending on which of the two half- spaces separated by that plane contains P. The y and z coordinates can be obtained in the same way from the xz- and xy-planes respectively. 54 CHAPTER 11. CARTESIAN COORDINATE SYSTEM

The coordinate surfaces of the Cartesian coordinates (x, y, z). The z-axis is vertical and the x-axis is highlighted in green. Thus, the red plane shows the points with x = 1, the blue plane shows the points with z = 1, and the yellow plane shows the points with y = −1. The three surfaces intersect at the point P (shown as a black sphere) with the Cartesian coordinates (1, −1, 1).

11.2.4 Higher dimensions

A Euclidean plane with a chosen Cartesian system is called a Cartesian plane. Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of real numbers; that is with the Cartesian product R2 = R × R , where R is the set of all reals. In the same way, the points any Euclidean space of dimension n be identified with the tuples (lists) of n real numbers, that is, with the Cartesian product Rn .

11.2.5 Generalizations

The concept of Cartesian coordinates generalizes to allow axes that are not perpendicular to each other, and/or dif- ferent units along each axis. In that case, each coordinate is obtained by projecting the point onto one axis along a direction that is parallel to the other axis (or, in general, to the hyperplane defined by all the other axes). In such an oblique coordinate system the computations of distances and angles must be modified from that in standard Cartesian systems, and many standard formulas (such as the Pythagorean formula for the distance) do not hold. 11.3. NOTATIONS AND CONVENTIONS 55

11.3 Notations and conventions

The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10, 5) or (3, 5, 7). The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters x and y on the plane, and x, y, and z in three-dimensional space. This custom comes from a convention of algebra, which use letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities. These conventional names are often used in other domains, such as physics and engineering, although other letters may be used. For example, in a graph showing how a pressure varies with time, the graph coordinates may be denoted t and p. Each axis is usually named after the coordinate which is measured along it; so one says the x-axis, the y-axis, the t-axis, etc.

Another common convention for coordinate naming is to use subscripts, as in x1, x2, ... xn for the n coordinates in an n-dimensional space; especially when n is greater than 3, or not specified. Some authors prefer the numbering x0, x1, ... xn₋₁. These notations are especially advantageous in computer programming: by storing the coordinates of a point as an array, instead of a record, the subscript can serve to index the coordinates. In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called the abscissa) is measured along a horizontal axis, oriented from left to right. The second coordinate (the ordinate) is then measured along a vertical axis, usually oriented from bottom to top. However, computer graphics and image processing often use a coordinate system with the y axis oriented downwards on the computer display. This convention developed in the 1960s (or earlier) from the way that images were originally stored in display buffers. For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z axis added to represent height (positive up). Furthermore, there is a convention to orient the x-axis toward the viewer, biased either to the right or left. If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically, respectively, then the z axis should be shown pointing “out of the page” towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this right- handedness, which ensures consistency. For 3D diagrams, the names “abscissa” and “ordinate” are rarely used for x and y, respectively. When they are, the z-coordinate is sometimes called the applicate. The words abscissa, ordinate and applicate are sometimes used to refer to coordinate axes rather than the coordinate values.[4]

11.3.1 Quadrants and octants

Main articles: Octant (solid geometry) and Quadrant (plane geometry) The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the two coordinates are +,+), II (−,+), III (−,−), and IV (+,−). When the axes are drawn according to the mathematical custom, the numbering goes counter-clockwise starting from the upper right (“north-east”) quadrant. Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or octants, according to the signs of the coordinates of the points. The convention used for naming a specific octant is to list its signs, e.g. (+ + +) or (− + −). The generalization of the quadrant and octant to an arbitrary number of dimensions is the orthant, and a similar naming system applies.

11.4 Cartesian formulae for the plane

11.4.1 Distance between two points

The Euclidean distance between two points of the plane with Cartesian coordinates (x1, y1) and (x2, y2) is 56 CHAPTER 11. CARTESIAN COORDINATE SYSTEM

The four quadrants of a Cartesian coordinate system.

√ 2 2 d = (x2 − x1) + (y2 − y1) . This is the Cartesian version of Pythagoras’s theorem. In three-dimensional space, the distance between points (x1, y1, z1) and (x2, y2, z2) is

√ 2 2 2 d = (x2 − x1) + (y2 − y1) + (z2 − z1) , which can be obtained by two consecutive applications of Pythagoras’ theorem.

11.4.2 Euclidean transformations

The Euclidean transformations or Euclidean motions are the (bijective) mappings of points of the Euclidean plane to themselves which preserve distances between points. There are four types of these mappings (also called isometries): translations, rotations, reflections and glide reflections.[5]

Translation

Translating a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers (a, b) to the Cartesian coordinates of every point in the set. That is, if the original coordinates 11.4. CARTESIAN FORMULAE FOR THE PLANE 57

of a point are (x, y), after the translation they will be

(x′, y′) = (x + a, y + b).

Rotation

To rotate a figure counterclockwise around the origin by some angle θ is equivalent to replacing every point with coordinates (x,y) by the point with coordinates (x',y'), where

x′ = x cos θ − y sin θ

y′ = x sin θ + y cos θ. Thus: (x′, y′) = ((x cos θ − y sin θ ), (x sin θ + y cos θ )).

Reflection

If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the Y-axis), as if that line were a mirror. Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the X-axis). In more generality, reflection across a line through the origin making an angle θ with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where

x′ = x cos 2θ + y sin 2θ

y′ = x sin 2θ − y cos 2θ. Thus: (x′, y′) = ((x cos 2θ + y sin 2θ ), (x sin 2θ − y cos 2θ )).

Glide reflection

A glide reflection is the composition of a reflection across a line followed by a translation in the direction of that line. It can be seen that the order of these operations does not matter (the translation can come first, followed by the reflection).

General matrix form of the transformations

These Euclidean transformations of the plane can all be described in a uniform way by using matrices. The result (x′, y′) of applying a Euclidean transformation to a point (x, y) is given by the formula

(x′, y′) = (x, y)A + b

[6] where A is a 2×2 orthogonal matrix and b = (b1, b2) is an arbitrary ordered pair of numbers; that is,

′ x = xA11 + yA21 + b1

′ y = xA12 + yA22 + b2, where ( ) A A A = 11 12 . [Note the use of row vectors for point coordinates and that the matrix A21 A22 is written on the right.] 58 CHAPTER 11. CARTESIAN COORDINATE SYSTEM

To be orthogonal, the matrix A must have orthogonal rows with same Euclidean length of one, that is,

A11A21 + A12A22 = 0 and

2 2 2 2 A11 + A12 = A21 + A22 = 1.

This is equivalent to saying that A times its transpose must be the identity matrix. If these conditions do not hold, the formula describes a more general affine transformation of the plane provided that the determinant of A is not zero. The formula defines a translation if and only if A is the identity matrix. The transformation is a rotation around some point if and only if A is a rotation matrix, meaning that

A11A22 − A21A12 = 1.

A reflection or glide reflection is obtained when,

A11A22 − A21A12 = −1.

Assuming that translation is not used transformations can be combined by simply multiplying the associated trans- formation matrices.

Affine transformation

Another way to represent coordinate transformations in Cartesian coordinates is through affine transformations. In affine transformations an extra dimension is added and all points are given a value of 1 for this extra dimension. The advantage of doing this is that point translations can be specified in the final column of matrix A. In this way, all of the euclidean transformations become transactable as matrix point multiplications. The affine transformation is given by:

     ′ A11 A21 b1 x x     ′ A12 A22 b2 y = y . [Note the matrix A from above was transposed. The 0 0 1 1 1 matrix is on the left and column vectors for point coordinates are used.]

Using affine transformations multiple different euclidean transformations including translation can be combined by simply multiplying the corresponding matrices.

Scaling

An example of an affine transformation which is not a Euclidean motion is given by scaling. To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If (x, y) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates

(x′, y′) = (mx, my).

If m is greater than 1, the figure becomes larger; if m is between 0 and 1, it becomes smaller. 11.5. ORIENTATION AND HANDEDNESS 59

Shearing

A shearing transformation will push the top of a square sideways to form a parallelogram. Horizontal shearing is defined by:

(x′, y′) = (x + ys, y)

Shearing can also be applied vertically:

(x′, y′) = (x, xs + y)

11.5 Orientation and handedness

Main article: Orientation (mathematics) See also: right-hand rule and Axes conventions

11.5.1 In two dimensions

The right hand rule.

Fixing or choosing the x-axis determines the y-axis up to direction. Namely, the y-axis is necessarily the perpendicular to the x-axis through the point marked 0 on the x-axis. But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative. Each of these two choices determines a different orientation (also called handedness) of the Cartesian plane. 60 CHAPTER 11. CARTESIAN COORDINATE SYSTEM

The usual way of orienting the axes, with the positive x-axis pointing right and the positive y-axis pointing up (and the x-axis being the “first” and the y-axis the “second” axis) is considered the positive or standard orientation, also called the right-handed orientation. A commonly used mnemonic for defining the positive orientation is the right hand rule. Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the x-axis to the y-axis, in a positively oriented coordinate system. The other way of orienting the axes is following the left hand rule, placing the left hand on the plane with the thumb pointing up. When pointing the thumb away from the origin along an axis towards positive, the curvature of the fingers indicates a positive rotation along that axis. Regardless of the rule used to orient the axes, rotating the coordinate system will preserve the orientation. Switching any two axes will reverse the orientation, but switching both will leave the orientation unchanged.

11.5.2 In three dimensions

Fig. 7 – The left-handed orientation is shown on the left, and the right-handed on the right.

Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line. The two possible coordinate systems which result are called 'right-handed' and 'left- handed'. The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive. The name derives from the right-hand rule. If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative directions of the x-, y-, and z-axes in a right-handed system. The thumb indicates the x-axis, the index finger the y-axis and the middle finger the z-axis. Conversely, if the same is done with the left hand, a left-handed system results. Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point towards the observer, whereas the “middle” axis is meant to point away from the observer. The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). Hence the red arrow passes in front of the z-axis. 11.5. ORIENTATION AND HANDEDNESS 61

z-axis

x = 0 plane y = 0 yz plane xz origin plane y-axis z = 0 plane xy plane

x-axis

Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes.

3D Cartesian Coordinate Handedness

Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as “flipping in and out” between a convex cube and a concave “corner”. This corresponds to the two possible orientations of the coordinate system. Seeing the figure as convex gives a left-handed coordinate system. Thus the “correct” way to view 62 CHAPTER 11. CARTESIAN COORDINATE SYSTEM

Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

11.6 Representing a vector in the standard basis

A point in space in a Cartesian coordinate system may also be represented by a position vector, which can be thought of as an arrow pointing from the origin of the coordinate system to the point.[7] If the coordinates represent spatial positions (displacements), it is common to represent the vector from the origin to the point of interest as r . In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as:

r = xi + yj ( ) ( ) 1 0 where i = , and j = are unit vectors in the direction of the x-axis and y-axis respectively, generally 0 1 referred to as the standard basis (in some application areas these may also be referred to as versors). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates (x, y, z) can be written as:[8]

r = xi + yj + zk   0 where k = 0 is the unit vector in the direction of the z-axis. 1 There is no natural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. In a two dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number z = x + iy. Here, i is the imaginary unit and is identified with the point with coordinates (0, 1), so it is not the unit vector in the direction of the x-axis. Since the complex numbers can be multiplied giving another complex number, this identification provides a means to “multiply” vectors. In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.

11.7 Applications

Cartesian coordinates are an abstraction that have a multitude of possible applications in the real world. However, three constructive steps are involved in superimposing coordinates on a problem application. 1) Units of distance must be decided defining the spatial size represented by the numbers used as coordinates. 2) An origin must be assigned to a specific spatial location or landmark, and 3) the orientation of the axes must be defined using available directional cues for (n-1) of the n axes. Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D). What units make sense? Kilometers are a good choice, since the original definition of the kilometer was geospa- tial...10,000 km equalling the surface distance from the Equator to the North Pole. Where to place the origin? Based on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite or- bits). Finally, how to orient X, Y and Z axis directions? The axis of Earth’s spin provides a natural direction strongly associated with “up vs. down”, so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the Prime Meridian stands out as a reference direction, so the X-axis takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions, and two perpendicular axes directions pinned down for X and Z, the Y-axis is determined by the first two choices. In order to obey the right hand rule, the Y-axis must point out from the geocenter to [ 90 degrees longitude, 0 degrees latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City? Using [ longitude = −73.985656, latitude = 40.748433 ], Earth radius = 40,000/2π, and transforming from spherical --> Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] = [ 1330.53 km, –4635.75 km, 4155.46 km ]. GPS navigation relies on such geocentric coordinates. In engineering projects, agreement on the definition of coordinates is a crucial foundation. One cannot assume that coordinates come predefined for a novel application, so knowledge of how to erect a coordinate system where there is none is essential to applying René Descartes’ ingenious thinking. 11.8. SEE ALSO 63

While spatial apps employ identical units along all axes, in business and scientific apps, each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). Although four- and higher- dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables. The graph of a function or relation is the set of all points satisfying that function or relation. For a function of one variable, f, the set of all points (x, y), where y = f(x) is the graph of the function f. For a function g of two variables, the set of all points (x, y, z), where z = g(x, y) is the graph of the function g. A sketch of the graph of such a function or relation would consist of all the salient parts of the function or relation which would include its relative extrema, its concavity and points of inflection, any points of discontinuity and its end behavior. All of these terms are more fully defined in calculus. Such graphs are useful in calculus to understand the nature and behavior of a function or relation.

11.8 See also

• Horizontal and vertical

• Jones diagram, which plots four variables rather than two.

• Orthogonal coordinates

• Polar coordinate system

• Spherical coordinate system

11.9 Notes

11.10 References

[1] “Analytic geometry”. Encyclopædia Britannica (Encyclopædia Britannica Online ed.). 2008.

[2] Burton 2011, p. 374

[3] A Tour of the Calculus, David Berlinski

[4] Springer online reference Encyclopedia of Mathematics

[5] Smart 1998, Chap. 2

[6] Brannan, Esplen & Gray 1998, pg. 49

[7] Brannan, Esplen & Gray 1998, Appendix 2, pp. 377–382

[8] David J. Griffiths (1999). Introduction to Electrodynamics. Prentice Hall. ISBN 0-13-805326-X.

11.11 Sources

• Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998), Geometry, Cambridge: Cambridge University Press, ISBN 0-521-59787-0

• Burton, David M. (2011), The History of Mathematics/An Introduction (7th ed.), New York: McGraw-Hill, ISBN 978-0-07-338315-6

• Smart, James R. (1998), Modern Geometries (5th ed.), Pacific Grove: Brooks/Cole, ISBN 0-534-35188-3 64 CHAPTER 11. CARTESIAN COORDINATE SYSTEM

11.12 Further reading

• Descartes, René (2001). Discourse on Method, Optics, Geometry, and Meteorology. Trans. by Paul J. Oscamp (Revised ed.). Indianapolis, IN: Hackett Publishing. ISBN 0-87220-567-3. OCLC 488633510. • Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers (1st ed.). New York: McGraw-Hill. pp. 55–79. LCCN 59-14456. OCLC 19959906.

• Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. LCCN 55-10911.

• Moon P, Spencer DE (1988). “Rectangular Coordinates (x, y, z)". Field Theory Handbook, Including Coordi- nate Systems, Differential Equations, and Their Solutions (corrected 2nd, 3rd print ed.). New York: Springer- Verlag. pp. 9–11 (Table 1.01). ISBN 978-0-387-18430-2. • Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. ISBN 0-07-043316-X. LCCN 52-11515. • Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. LCCN 67- 25285.

11.13 External links

• Cartesian Coordinate System

• Printable Cartesian Coordinates • Cartesian coordinates at PlanetMath.org.

• MathWorld description of Cartesian coordinates • Coordinate Converter – converts between polar, Cartesian and spherical coordinates

• Coordinates of a point Interactive tool to explore coordinates of a point

• open source JavaScript class for 2D/3D Cartesian coordinate system manipulation Chapter 12

Collinearity

Not to be confused with colinear map or multicollinearity.

In geometry, collinearity is a property of a set of points, specifically, the property of lying on a single line.[1] A set of points with this property is said to be collinear (sometimes spelled as colinear[2]). In greater generality, the term has been used for aligned objects, that is, things being “in a line” or “in a row”.

12.1 Points on a line

In any geometry, the set of points on a line are said to be collinear. In Euclidean geometry this relation is intuitively visualized by points lying in a row on a “straight line”. However, in most geometries (including Euclidean) a line is typically a primitive (undefined) object type, so such visualizations will not necessarily be appropriate. A model for the geometry offers an interpretation of how the points, lines and other object types relate to one another and a notion such as collinearity must be interpreted within the context of that model. For instance, in spherical geometry, where lines are represented in the standard model by great circles of a sphere, sets of collinear points lie on the same great circle. Such points do not lie on a “straight line” in the Euclidean sense, and are not thought of as being in a row. A mapping of a geometry to itself which sends lines to lines is called a collineation, it preserves the collinearity property. The linear maps (or linear functions) of vector spaces, viewed as geometric maps, map lines to lines, that is, they map collinear point sets to collinear point sets and so, are collineations. In projective geometry these linear mappings are called homographies and are just one type of collineation.

12.2 Examples in Euclidean geometry

12.2.1 Triangles

In any triangle the following sets of points are collinear:

• The orthocenter, the circumcenter, the centroid, the Exeter point, the de Longchamps point, and the center of the nine-point circle are collinear, all falling on a line called the Euler line.

• The de Longchamps point also has other collinearities.

• Any vertex, the tangency of the opposite side with an excircle, and the Nagel point are collinear in a line called a splitter of the triangle.

• The midpoint of any side, the point that is equidistant from it along the triangle’s boundary in either direction (so these two points bisect the perimeter), and the center of the Spieker circle are collinear in a line called a cleaver of the triangle. (The Spieker circle is the incircle of the medial triangle, and its center is the center of mass of the perimeter of the triangle.)

65 66 CHAPTER 12. COLLINEARITY

• Any vertex, the tangency of the opposite side with the incircle, and the Gergonne point are collinear.

• From any point on the circumcircle of a triangle, the nearest points on each of the three extended sides of the triangle are collinear in the Simson line of the point on the circumcircle.

• The lines connecting the feet of the altitudes intersect the opposite sides at collinear points.[3]:p.199

• A triangle’s incenter, the midpoint of an altitude, and the point of contact of the corresponding side with the excircle relative to that side are collinear.[4]:p.120,#78

• Menelaus’ theorem states that three points P1,P2,P3 on the sides (some extended) of a triangle opposite vertices A1,A2,A3 respectively are collinear if and only if the following products of segment lengths are equal:[3]:p. 147

P1A2 · P2A3 · P3A1 = P1A3 · P2A1 · P3A2.

12.2.2 Quadrilaterals

• In a convex quadrilateral ABCD whose opposite sides intersect at E and F, the midpoints of AC, BD, and EF are collinear and the line through them is called the Newton line (sometimes known as the Newton-Gauss line). If the quadrilateral is a tangential quadrilateral, then its incenter also lies on this line.[5]

• In a convex quadrilateral, the quasiorthocenter H, the “area centroid” G, and the quasicircumcenter O are collinear in this order, and HG = 2GO.[6] (See Quadrilateral#Remarkable points and lines in a convex quadri- lateral.)

• Other collinearities of a tangential quadrilateral are given in Tangential quadrilateral#Collinear_points.

• In a cyclic quadrilateral, the circumcenter, the vertex centroid (the intersection of the two bimedians), and the anticenter are collinear.[7]

• In a cyclic quadrilateral, the area centroid, the vertex centroid, and the intersection of the diagonals are collinear.[8]

• In a tangential trapezoid, the tangencies of the incircle with the two bases are collinear with the incenter.

• In a tangential trapezoid, the midpoints of the legs are collinear with the incenter.

12.2.3 Hexagons

• Pascal’s theorem (also known as the Hexagrammum Mysticum Theorem) states that if an arbitrary six points are chosen on a conic section (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon. The converse is also true: the Braikenridge– Maclaurin theorem states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line, then the six vertices of the hexagon lie on a conic, which may be degenerate as in Pappus’s hexagon theorem.

12.2.4 Conic sections

• By Monge’s theorem, for any three circles in a plane, none of which is inside one of the others, the three intersection points of the three pairs of lines, each externally tangent to two of the circles, are collinear.

• In an ellipse, the center, the two foci, and the two vertices with the smallest radius of curvature are collinear, and the center and the two vertices with the greatest radius of curvature are collinear.

• In a hyperbola, the center, the two foci, and the two vertices are collinear. 12.3. ALGEBRA 67

12.2.5 Cones

• The center of mass of a conic solid of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.

12.2.6 Tetrahedrons

• The centroid of a tetrahedron is the midpoint between its Monge point and circumcenter. These points define the Euler line of the tetrahedron that is analogous to the Euler line of a triangle. The center of the tetrahedron’s twelve-point sphere also lies on the Euler line.

12.3 Algebra

12.3.1 Collinearity of points whose coordinates are given

In coordinate geometry, in n-dimensional space, a set of three or more distinct points are collinear if and only if, the matrix of the coordinates of these vectors is of rank 1 or less. For example, given three points X = (x1, x2, ... , xn), Y = (y1, y2, ... , yn), and Z = (z1, z2, ... , zn), if the matrix

  x1 x2 . . . xn   y1 y2 . . . yn z1 z2 . . . zn is of rank 1 or less, the points are collinear.

Equivalently, for every subset of three points X = (x1, x2, ... , xn), Y = (y1, y2, ... , yn), and Z = (z1, z2, ... , zn), if the matrix

  1 x1 x2 . . . xn   1 y1 y2 . . . yn 1 z1 z2 . . . zn is of rank 2 or less, the points are collinear. In particular, for three points in the plane (n = 2), the above matrix is square and the points are collinear if and only if its determinant is zero; since that 3 × 3 determinant is plus or minus twice the area of a triangle with those three points as vertices, this is equivalent to the statement that the three points are collinear if and only if the triangle with those points as vertices has zero area.

12.3.2 Collinearity of points whose pairwise distances are given

A set of at least three distinct points is called straight, meaning all the points are collinear, if and only if, for every three points A, B, and C, the following determinant of a Cayley–Menger determinant is zero (with d(AB) meaning the distance between A and B, etc.):

  0 d(AB)2 d(AC)2 1 d(AB)2 0 d(BC)2 1 det   = 0. d(AC)2 d(BC)2 0 1 1 1 1 0

This determinant is, by Heron’s formula, equal to −16 times the square of the area of a triangle with side lengths d(AB), d(BC), and d(AC); so checking if this determinant equals zero is equivalent to checking whether the triangle with vertices A, B, and C has zero area (so the vertices are collinear). Equivalently, a set of at least three distinct points are collinear if and only if, for every three points A, B, and C with d(AC) greater than or equal to each of d(AB) and d(BC), the triangle inequality d(AC) ≤ d(AB) + d(BC) holds with equality. 68 CHAPTER 12. COLLINEARITY

12.4 Number theory

Two numbers m and n are not coprime—that is, they share a common factor other than 1—if and only if for a rectangle plotted on a square lattice with vertices at (0, 0), (m, 0), (m, n), and (0, n), at least one interior point is collinear with (0, 0) and (m, n).

12.5 Concurrency (plane dual)

In various plane geometries the notion of interchanging the roles of “points” and “lines” while preserving the rela- tionship between them is called plane duality. Given a set of collinear points, by plane duality we obtain a set of lines all of which meet at a common point. The property that this set of lines has (meeting at a common point) is called concurrency, and the lines are said to be concurrent lines. Thus, concurrency is the plane dual notion to collinearity.

12.6 Collinearity graph

Given a partial geometry P, where two points determine at most one line, a collinearity graph of P is a graph whose vertices are the points of P, where two vertices are adjacent if and only if they determine a line in P.

12.7 Usage in statistics and econometrics

Main article: Multicollinearity

In statistics, collinearity refers to a linear relationship between two explanatory variables. Two variables are perfectly collinear if there is an exact linear relationship between the two, so the correlation between them is equal to 1 or −1. That is, X1 and X2 are perfectly collinear if there exist parameters λ0 and λ1 such that, for all observations i, we have

X2i = λ0 + λ1X1i.

This means that if the various observations (X₁i, X₂i ) are plotted in the (X1, X2) plane, these points are collinear in the sense defined earlier in this article. Perfect multicollinearity refers to a situation in which k (k ≥ 2) explanatory variables in a multiple regression model are perfectly linearly related, according to

Xki = λ0 + λ1X1i + λ2X2i + ··· + λk−1X(k−1),i for all observations i. In practice, we rarely face perfect multicollinearity in a data set. More commonly, the issue of multicollinearity arises when there is a “strong linear relationship” among two or more independent variables, meaning that

Xki = λ0 + λ1X1i + λ2X2i + ··· + λk−1X(k−1),i + εi where the variance of εi is relatively small. The concept of lateral collinearity expands on this traditional view, and refers to collinearity between explanatory and criteria (i.e., explained) variables.[9]

12.8 Usage in other areas 12.8. USAGE IN OTHER AREAS 69

An antenna mast with four collinear directional arrays.

12.8.1 Antenna arrays

In telecommunications, a collinear (or co-linear) antenna array is an array of dipole antennas mounted in such a manner that the corresponding elements of each antenna are parallel and aligned, that is they are located along a common line or axis. 70 CHAPTER 12. COLLINEARITY

12.8.2 Photography

The collinearity equations are a set of two equations, used in photogrammetry and remote sensing to relate coordinates in an image (sensor) plane (in two dimensions) to object coordinates (in three dimensions). In the photography setting, the equations are derived by considering the central projection of a point of the object through the optical centre of the camera to the image in the image (sensor) plane. The three points, object point, image point and optical centre, are always collinear. Another way to say this is that the line segments joining the object points with their image points are all concurrent at the optical centre.[10]

12.9 See also

• Pappus’s hexagon theorem • No-three-in-line problem

• Incidence (geometry)#Collinearity • Coplanarity

12.10 Notes

[1] The concept applies in any geometry Dembowski (1968, pg. 26), but is often only defined within the discussion of a specific geometry Coxeter (1969, pg. 168), Brannan, Esplen & Gray (1998, pg.106)

[2] Colinear (Merriam-Webster dictionary)

[3] Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929).

[4] Altshiller-Court, Nathan. College Geometry, Dover Publications, 1980.

[5] Dušan Djukić, Vladimir Janković, Ivan Matić, Nikola Petrović, The IMO Compendium, Springer, 2006, p. 15.

[6] Myakishev, Alexei (2006), “On Two Remarkable Lines Related to a Quadrilateral” (PDF), Forum Geometricorum 6: 289– 295.

[7] Honsberger, Ross (1995), “4.2 Cyclic quadrilaterals”, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, New Mathematical Library 37, Cambridge University Press, pp. 35–39, ISBN 978-0-88385-639-0

[8] Bradley, Christopher (2011), Three Centroids created by a Cyclic Quadrilateral (PDF)

[9] Kock, N.; Lynn, G. S. (2012). “Lateral collinearity and misleading results in variance-based SEM: An illustration and recommendations” (PDF). Journal of the Association for Information Systems 13 (7): 546–580.

[10] It’s more mathematically natural to refer to these equations as concurrency equations, but photogrammetry literature does not use that terminology.

12.11 References

• Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998), Geometry, Cambridge: Cambridge University Press, ISBN 0-521-59787-0

• Coxeter, H. S. M. (1969), Introduction to Geometry, New York: John Wiley & Sons, ISBN 0-471-50458-0 • Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275 Chapter 13

Endianness

Endianness refers to the order of the bytes, comprising a digital word, in computer memory. It also describes the order of byte transmission over a digital link. Words may be represented in big-endian or little-endian format. With big-endian the most-significant byte of a word is stored at a particular memory address and the subsequent bytes are stored in the following higher memory addresses, the least significant byte thus being stored at the highest memory address. Little-endian format reverses the order and stores the least-significant byte at the lower memory address with the most significant byte being stored at the highest memory address.[1] Both forms of endianness are widely used in digital electronics. The choice of endianness for a new design is often arbitrary, but later technology revisions and updates perpetuate the existing endianness and many other design at- tributes to maintain backward compatibility. As examples, the IBM z/Architecture mainframes use big-endian while the Intel x86 processors use little-endian. The designers chose endianness in the 1960s and 1970s respectively. Big-endian is the most common format in data networking; fields in the protocols of the Internet protocol suite, such as IPv4, IPv6, TCP, and UDP, are transmitted in big-endian order. For this reason, big-endian byte order is also referred to as network byte order. Little-endian storage is popular for microprocessors, in part due to significant influence on microprocessor designs by Intel Corporation. Mixed forms also exist, for instance the ordering of bytes in a 16-bit word may differ from the ordering of 16-bit words within a 32-bit word. Such cases are sometimes referred to as mixed-endian or middle-endian. There are also some bi-endian processors that operate in either little-endian or big-endian mode.

13.1 Illustration

Endianness may be demonstrated by writing a decimal number on paper in the conventional way which uses positional notation. The digits are written left to right, with the most significant digit to the left, which is analogous to the lowest address of memory used to store a number. For example, the number 123, has the hundreds-digit, 1, left-most which is understood by a numerate reader. This is an example of a big-endian convention taken from daily life. The little-endian way of writing the same number would place the digit 1 in the right-most position: 321. A person following conventional place-value order, who is not aware of this special ordering, would read the number as three hundred and twenty one. Endianness in computing is similar, but it applies to the ordering of bytes in memory or during transmission. The illustrations to the right, where a is a memory address, show big-endian and little-endian storage in memory.

13.2 History

Before microprocessors, most computers used big-endian, but the PDP-11 was a notable exception. The Datapoint 2200 used simple bit-serial logic with little-endian to facilitate carry propagation. When Intel developed the 8008 microprocessor for Datapoint, they used little-endian for compatibility. However, as Intel was unable to deliver the 8008 in time, Datapoint used a medium scale integration (MSI) equivalent.[2][3] Dealing with data of different endianness is sometimes termed the NUXI problem.[4] This terminology alludes to

71 72 CHAPTER 13. ENDIANNESS

32-bit integer

Memory 0A0B0C0D ......

a: 0A a+1: 0B a+2: 0C a+3: 0D Big-endian

Big-Endian

the byte order conflicts encountered while adapting UNIX, which ran on the little-endian PDP-11, to a big-endian computer such as the IBM Series/1. One of the first programs converted was supposed to print out Unix, but on the Series/1 it printed nUxi instead.[5] Unix was one of the first systems to allow the same code to run on, and transfer data between, platforms with different internal representations.

13.2.1 Etymology

Danny Cohen introduced use of the terms Little-Endian and Big-Endian for byte ordering in a well-known document in 1980.[6][7] In this technical and political examination of byte ordering issues, the “endian” names were pointedly drawn from Jonathan Swift's 1726 satirical fantasy novel, Gulliver’s Travels, in which civil war erupts over whether the big or the small end of a soft-boiled egg is the proper end to crack open.[8][9]

13.3 Hardware

Computer memory consists of a sequence of storage cells. Each cell is identified in hardware and software by its memory address. If the total number of storage cells in memory is n, then addresses are enumerated from 0 to n-1. Computer programs often use data structures of fields that may consist of more data than is stored in one memory cell. For the purpose of this article where its use as an operand of an instruction is relevant, a field consists of a consecutive sequence of bytes and represents a simple data value. In addition to that, it has to be of numeric type in some positional number system (mostly base-10 or base-2 — or base-256 in case of 8-bit bytes).[10] In such a number 13.3. HARDWARE 73

32-bit integer

0A0B0C0D Memory ......

a: 0D a+1: 0C a+2: 0B a+3: 0A Little-endian

Little-Endian

system the “value” of a digit is determined not only by its value as a single digit, but also by the position it holds in the complete number, its “significance”. These positions can be mapped to memory mainly in two ways:[11]

• increasing numeric significance with increasing memory addresses (or increasing time), known as little-endian, and

• decreasing numeric significance with increasing memory addresses (or increasing time), known as big-endian[12]

The Intel x86 and x86-64 series of processors use the little-endian format, and for this reason, the little-endian format is also known in the industry as the "Intel convention".[13][14] Other well-known little-endian processor architectures are the 6502 (including 65802, 65C816), Z80 (including Z180, eZ80 etc.), MCS-48, DEC Alpha, Altera Nios II, Atmel AVR, VAX, and, largely, PDP-11. The Intel 8051, contrary to other Intel processors, expects 16-bit addresses in big-endian format, except for the LCALL instruction whose target address is stored in little-endian format.[15] The Motorola 6800 and 68k series of processors use the big-endian format, and for this reason, the big-endian for- mat is also known as the "Motorola convention".[13][14] Other well-known processors that use the big-endian format include the Xilinx Microblaze, SuperH, IBM POWER, Atmel AVR32, and System/360 and its successors such as System/370, ESA/390, and z/Architecture. The PDP-10 also used big-endian addressing for byte-oriented instruc- tions. SPARC historically used big-endian until version 9, which is bi-endian, similarly the ARM architecture was little- endian before version 3 when it became bi-endian, and the PowerPC and Power Architecture descendants of POWER are also bi-endian. 74 CHAPTER 13. ENDIANNESS

13.3.1 Bi-endian hardware

Some architectures (including ARM versions 3 and above, PowerPC, Alpha, SPARC V9, MIPS, PA-RISC, SuperH SH-4 and IA-64) feature a setting which allows for switchable endianness in data segments, code segments or both. This feature can improve performance or simplify the logic of networking devices and software. The word bi-endian, when said of hardware, denotes the capability of the machine to compute or pass data in either endian format. Many of these architectures can be switched via software to default to a specific endian format (usually done when the computer starts up); however, on some systems the default endianness is selected by hardware on the motherboard and cannot be changed via software (e.g. the Alpha, which runs only in big-endian mode on the Cray T3E). Note that the term “bi-endian” refers primarily to how a processor treats data accesses. Instruction accesses (fetches of instruction words) on a given processor may still assume a fixed endianness, even if data accesses are fully bi-endian, though this is not always the case, such as on Intel’s IA-64-based Itanium CPU, which allows both. Note, too, that some nominally bi-endian CPUs require motherboard help to fully switch endianness. For instance, the 32-bit desktop-oriented PowerPC processors in little-endian mode act as little-endian from the point of view of the executing programs, but they require the motherboard to perform a 64-bit swap across all 8 byte lanes to ensure that the little-endian view of things will apply to I/O devices. In the absence of this unusual motherboard hardware, device driver software must write to different addresses to undo the incomplete transformation and also must perform a normal byte swap. Some CPUs, such as many PowerPC processors intended for embedded use, allow per-page choice of endianness.

13.3.2 Floating-point

Although the ubiquitous x86 processors of today use little-endian storage for all types of data (integer, floating point, BCD), there have been a few historical machines where floating point numbers were represented in big-endian form while integers were represented in little-endian form.[16] There are old ARM processors that have half little-endian, half big-endian floating point representation for double-precision numbers: both 32-bit words are stored in little- endian like integer registers, but the most significant one first. Because there have been many floating point formats with no "network" standard representation for them, there is no formal standard for transferring floating point values between diverse systems. It may therefore appear strange that the widespread IEEE 754 floating point standard does not specify endianness.[17] Theoretically, this means that even standard IEEE floating point data written by one machine might not be readable by another. However, on modern standard computers (i.e., implementing IEEE 754), one may in practice safely assume that the endianness is the same for floating point numbers as for integers, making the conversion straightforward regardless of data type. (Small embedded systems using special floating point formats may be another matter however.)

13.4 Optimization

The little-endian system has the property that the same value can be read from memory at different lengths without using different addresses (even when alignment restrictions are imposed). For example, a 32-bit memory location with content 4A 00 00 00 can be read at the same address as either 8-bit (value = 4A), 16-bit (004A), 24-bit (00004A), or 32-bit (0000004A), all of which retain the same numeric value. Although this little-endian property is rarely used directly by high-level programmers, it is often employed by code optimizers as well as by assembly language programmers. On the other hand, in some situations it may be useful to obtain an approximation of a multi-byte or multi-word value by reading only its most-significant portion instead of the complete representation; a big-endian processor may read such an approximation using the same base-address that would be used for the full value.

13.4.1 Calculation order

Little-endian representation simplifies hardware in processors that add multi-byte integral values a byte at a time, such as small-scale byte-addressable processors and microcontrollers. As carry propagation must start at the least significant bit (and thus byte), multi-byte addition can then be carried out with a monotonically-incrementing address sequence, a simple operation already present in hardware. On a big-endian processor, its addressing unit has to be 13.5. MAPPING MULTI-BYTE BINARY VALUES TO MEMORY 75

told how big the addition is going to be so that it can hop forward to the least significant byte, then count back down towards the most significant. However, high-performance processors usually perform these operations simultaneously, fetching multi-byte operands from memory as a single operation, so that the complexity of the hardware is not affected by the byte ordering.

13.5 Mapping multi-byte binary values to memory

Let us agree to understand the orientation left to right in memory as increasing memory addresses − as in the table to the left. Furthermore, the hex value 0x0a0b0c0d is defined to be the value 168496141 in the usual (and big-endian style) decimal notation. If you map this value as a binary value to a sequence of 4 bytes in memory in big-endian style, you are writing the bytes from left to right in decreasing significance: 0A at +0, 0B at +1, 0C at +2, 0D at +3. However, on a little-endian system, the bytes are written from left to right in increasing significance, starting with the one’s byte: 0D at +0, 0C at +1, 0B at +2, 0A at +3. If you write a 32-bit binary value to a memory location on a little-endian system and after that output the memory location (with growing addresses from left to right), then the output of the memory is reversed (byte-swapped) compared to usual big-endian notation. This is the way a hexdump is displayed: because the dumping program is unable to know what kind of data it is dumping, the only orientation it can observe is monotonically increasing addresses. The human reader, however, who knows that he is reading a hexdump of a little-endian system and who knows what kind of data he is reading, reads the byte sequence 0D,0C,0B,0A as the 32-bit binary value 168496141, or 0x0a0b0c0d in hexadecimal notation. (Of course, this is not the same as the number 0D0C0B0A = 0x0d0c0b0a = 218893066.)

13.6 Examples

This section provides example layouts of the 32-bit number 0A0B0C0D in the most common variants of endianness. There exist several digital processors that use other formats. That is true for typical embedded systems as well as for general computer CPUs. Most processors used in non CPU roles in typical computers (in storage units, peripherals etc.) also use one of these two basic formats, although not always 32-bit. The examples refer to the storage in memory of the value. It uses hexadecimal notation.

13.6.1 Big-endian

Atomic element size 8-bit

address increment 1-byte (octet)

The most significant byte (MSB) value, which is 0A in our example, is stored at the memory location with the lowest address, the next byte value in significance, 0B, is stored at the following memory location and so on. This is akin to left-to-right reading in hexadecimal order.

Atomic element size 16-bit

The most significant atomic element stores now the value 0A0B, followed by 0C0D.

13.6.2 Little-endian

Atomic element size 8-bit

address increment 1-byte (octet)

The least significant byte (LSB) value, 0D, is at the lowest address. The other bytes follow in increasing order of significance. 76 CHAPTER 13. ENDIANNESS

32-bit integer

Memory 0A0B0C0D ......

a: 0A a+1: 0B a+2: 0C a+3: 0D Big-endian

Atomic element size 16-bit

The least significant 16-bit unit stores the value 0C0D, immediately followed by 0A0B. Note that 0C0D and 0A0B represent integers, not bit layouts.

When organized by byte addresses

Byte addresses increasing from right to left

Visualising memory addresses from left to right makes little-endian values appear backwards. If the addresses are written increasing towards the left instead, each individual little-endian value will appear forwards. However strings of values or characters appear reversed instead. With 8-bit atomic elements: The least significant byte (LSB) value, 0D, is at the lowest address. The other bytes follow in increasing order of significance. With 16-bit atomic elements: The least significant 16-bit unit stores the value 0C0D, immediately followed by 0A0B. The display of text is reversed from the normal display of languages such as English that read from left to right. For example, the word “XRAY” displayed in this manner, with each character stored in an 8-bit atomic element: If pairs of characters are stored in 16-bit atomic elements (using 8 bits per character), it could look even stranger: 13.6. EXAMPLES 77

32-bit integer

0A0B0C0D Memory ......

a: 0D a+1: 0C a+2: 0B a+3: 0A Little-endian

This conflict between the memory arrangements of binary data and text is intrinsic to the nature of the little-endian convention, but is a conflict only for languages written left-to-right, such as English. For right-to-left languages such as Arabic and Hebrew, there is no conflict of text with binary, and the preferred display in both cases would be with addresses increasing to the left. (On the other hand, right-to-left languages have a complementary intrinsic conflict in the big-endian system.)

13.6.3 Middle-endian

Numerous other orderings, generically called middle-endian or mixed-endian, are possible. On the PDP-11 (16- bit little-endian) for example, the compiler stored 32-bit values with the 16-bit halves swapped from the expected little-endian order. This ordering is known as PDP-endian.

• storage of a 32-bit word (hexadecimal 0A0B0C0D) on a PDP-11

The ARM architecture can also produce this format when writing a 32-bit word to an address 2 bytes from a 32-bit word alignment. Segment descriptors on Intel 80386 and compatible processors keep a base 32-bit address of the segment stored in little-endian order, but in four nonconsecutive bytes, at relative positions 2, 3, 4 and 7 of the descriptor start. 78 CHAPTER 13. ENDIANNESS

13.7 Networking

Many IETF RFCs use the term network order, meaning the order of transmission for bits and bytes over the wire in network protocols. Among others, the historic RFC 1700 (also known as Internet standard STD 2) has defined the network order for protocols in the Internet protocol suite to be big-endian, hence the use of the term “network byte order” for big-endian byte order; however, not all protocols use big-endian byte order as the network order.[18] The Berkeley sockets API defines a set of functions to convert 16-bit and 32-bit integers to and from network byte order: the htons (host-to-network-short) and htonl (host-to-network-long) functions convert 16-bit and 32-bit values respectively from machine (host) to network order; the ntohs and ntohl functions convert from network to host order. These functions may be a no-op on a big-endian system. The telephone network has always sent the most significant part first, the area code.[19] In CANopen, multi-byte parameters are always sent least significant byte first (little endian). The same is true for Ethernet Powerlink.[20] While the lowest network protocols may deal with sub-byte formatting, all the layers above them usually consider the byte (mostly meant as octet) as their atomic unit.

13.8 Files and byte swap

Endianness is a problem when a binary file created on a computer is read on another computer with different en- dianness. Some compilers have built-in facilities to deal with data written in other formats. For example, the Intel Fortran compiler supports the non-standard CONVERT specifier, so a file can be opened as

OPEN(unit,CONVERT='BIG_ENDIAN',...) or

OPEN(unit,CONVERT='LITTLE_ENDIAN',...)

Some compilers have options to generate code that globally enables the conversion for all file IO operations. This allows programmers to reuse code on a system with the opposite endianness without having to modify the code itself. If the compiler does not support such conversion, the programmer needs to swap the bytes via ad hoc code. Fortran sequential unformatted files created with one endianness usually cannot be read on a system using the other endianness because Fortran usually implements a record (defined as the data written by a single Fortran statement) as data preceded and succeeded by count fields, which are integers equal to the number of bytes in the data. An attempt to read such file on a system of the other endianness then results in a run-time error, because the count fields are incorrect. This problem can be avoided by writing out sequential binary files as opposed to sequential unformatted. Unicode text can optionally start with a byte order mark (BOM) to signal the endianness of the file or stream. Its code point is U+FEFF. In UTF-32 for example, a big-endian file should start with 00 00 FE FF; a little-endian should start with FF FE 00 00. Application binary data formats, such as for example MATLAB .mat files, or the .BIL data format, used in topography, are usually endianness-independent. This is achieved by:

1. storing the data always in one fixed endianness, or

2. carrying with the data a switch to indicate which endianness the data was written with.

When reading the file, the application converts the endianness, invisibly from the user. An example of the first case is the binary XLS file format that is portable between Windows and Mac systems and always little endian, leaving the Mac application to swap the bytes on load and save.[21] TIFF image files are an example of the second strategy, whose header instructs the application about endianness of their internal binary integers. If a file starts with the signature “MM” it means that integers are represented as big- endian, while “II” means little-endian. Those signatures need a single 16-bit word each, and they are palindromes 13.9. BIT ENDIANNESS 79

(that is, they read the same forwards and backwards), so they are endianness independent. “I” stands for Intel and “M” stands for Motorola, the respective CPU providers of the IBM PC compatibles (Intel) and Apple Macintosh platforms (Motorola) in the 1980s. Intel CPUs are little-endian, while Motorola 680x0 CPUs are big-endian. This explicit signature allows a TIFF reader program to swap bytes if necessary when a given file was generated by a TIFF writer program running on a computer with a different endianness. Note that since the required byte swap depends on the size of the numbers stored in the file (two 2-byte integers require a different swap than one 4-byte integer), the file format must be known to perform endianness conversion. /* C function to change endianness for byte swap in an unsigned 32-bit integer */ uint32_t ChangeEndianness(uint32_t value) { uint32_t result = 0; result |= (value & 0x000000FF) << 24; result |= (value & 0x0000FF00) << 8; result |= (value & 0x00FF0000) >> 8; result |= (value & 0xFF000000) >> 24; return result; }

13.9 Bit endianness

Bit numbering is a similar concept to endianness in bit-level systems. Bit or bit-level endianness refers to the trans- mission order of bits over a serial medium. Usually that order is transparently managed by the hardware and is the bit-level analogue of little-endian (low-bit first), as in RS-232, Ethernet, and USB. Some protocols use the opposite ordering (e.g. Teletext, I²C, and SONET and SDH[22]). As bit ordering is usually only relevant on a very low level, terms like LSB first and MSB first are more descriptive for bit order than the concept of endianness. The terms bit endianness and bit-level endianness are seldom used when talking about the representation of a stored value, as they are only meaningful for the rare computer architectures where each individual bit has a unique address.

13.10 References

[1] University of Maryland - Definitions. Accessed 26 Sept 2014 [2] House, David; Fagin, Federico; Feeney, Hal; Gelbach, Ed; Hoff, Ted; Mazor, Stan; Smith, Hank (2006-09-21). “Oral History Panel on the Development and Promotion of the Intel 8008 Microprocessor” (PDF). Computer History Museum. p. 5. Retrieved 23 April 2014. Mazor: And lastly, the original design for Datapoint... what they wanted was a [bit] serial machine. And if you think about a serial machine, you have to process all the addresses and data one-bit at a time, and the rational way to do that is: low-bit to high-bit because that’s the way that carry would propagate. So it means that [in] the jump instruction itself, the way the 14-bit address would be put in a serial machine is bit-backwards, as you look at it, because that’s the way you’d want to process it. Well, we were gonna built a byte-parallel machine, not bit-serial and our compromise (in the spirit of the customer and just for him), we put the bytes in backwards. We put the low- byte [first] and then the high-byte. This has since been dubbed “Little Endian” format and it’s sort of contrary to what you’d think would be natural. Well, we did it for Datapoint. As you’ll see, they never did use the [8008] chip and so it was in some sense “a mistake”, but that [Little Endian format] has lived on to the 8080 and 8086 and [is] one of the marks of this family. [3] Ken Lunde (13 January 2009). CJKV Information Processing. O'Reilly Media, Inc. p. 29. ISBN 978-0-596-51447-1. Retrieved 21 May 2013. [4] “NUXI problem”. The Jargon File. Retrieved 2008-12-20. [5] Jalics, Paul J.; Heines, Thomas S. (1 December 1983). “Transporting a portable operating system: UNIX to an IBM minicomputer”. Communications of the ACM 26 (12): 1066–1072. doi:10.1145/358476.358504. [6] Danny Cohen (1980-04-01). On Holy Wars and a Plea for Peace. IETF. IEN 137. http://www.ietf.org/rfc/ien/ien137.txt. "…which bit should travel first, the bit from the little end of the word, or the bit from the big end of the word? The followers of the former approach are called the Little-Endians, and the followers of the latter are called the Big-Endians.” Also published at IEEE Computer, October 1981 issue. [7] “Internet Hall of Fame Pioneer”. Internet Hall of Fame. The Internet Society. [8] Jonathan Swift (1726). Gulliver’s Travels. [9] David Cary. “Endian FAQ”. Retrieved 2010-10-11. [10] When character (text) strings are compared with one another, this is done lexicographically where a single positional element (character) also has a positional value. Lexicographical comparison means almost everywhere: first character ranks highest — as in the telephone book. This would have the consequence that almost every machine would be big-endian or at least mixed-endian. Therefore, for the criterion below to apply, the data type in question has to be numeric. 80 CHAPTER 13. ENDIANNESS

[11] Andrew S. Tanenbaum; Todd M. Austin (4 August 2012). Structured Computer Organization. Prentice Hall PTR. ISBN 978-0-13-291652-3. Retrieved 18 May 2013.

[12] Note that, in these expressions, the term “end” is meant as “extremity”, not as “last part"; and that (the extremity with) big resp. little significance is written first.

[13] Küveler, Gerd; Schwoch, Dietrich (2013) [1996]. Arbeitsbuch Informatik - eine praxisorientierte Einführung in die Daten- verarbeitung mit Projektaufgabe (in German). Vieweg-Verlag, reprint: Springer-Verlag. doi:10.1007/978-3-322-92907-5. ISBN 978-3-528-04952-2. 9783322929075. Retrieved 2015-08-05.

[14] Küveler, Gerd; Schwoch, Dietrich (2007-10-04). Informatik für Ingenieure und Naturwissenschaftler: PC- und Mikrocom- putertechnik, Rechnernetze (in German) 2 (5 ed.). Vieweg, reprint: Springer-Verlag. ISBN 3834891916. 9783834891914. Retrieved 2015-08-05.

[15] http://www.keil.com/support/man/docs/c51/c51_xe.htm

[16] “Floating point formats”.

[17] “pack – convert a list into a binary representation”.

[18] Reynolds, J.; Postel, J. (October 1994). “Data Notations”. Assigned Numbers. IETF. p. 3. STD 2. RFC 1700. https: //tools.ietf.org/html/rfc1700#page-3. Retrieved 2012-03-02.

[19] “Webster’s New World Telecom Dictionary”.

[20] Ethernet POWERLINK Standardisation Group (2012), EPSG Working Draft Proposal 301: Ethernet POWERLINK Com- munication Profile Specification Version 1.1.4, chapter 6.1.1.

[21] “Microsoft Office Excel 97 - 2007 Binary File Format Specification (*.xls 97-2007 format)". Microsoft Corporation. 2007.

[22] Cf. Sec. 2.1 Bit Transmission of http://tools.ietf.org/html/draft-ietf-pppext-sonet-as-00

13.11 Further reading

• Danny Cohen (1980-04-01). On Holy Wars and a Plea for Peace. IETF. IEN 137. http://www.ietf.org/rfc/ ien/ien137.txt. Also published at IEEE Computer, October 1981 issue. • David V. James (June 1990). “Multiplexed buses: the endian wars continue”. IEEE Micro 10 (3): 9–21. doi:10.1109/40.56322. ISSN 0272-1732. Retrieved 2008-12-20. • Bertrand Blanc, Bob Maaraoui (December 2005). “Endianness or Where is Byte 0?" (PDF). Retrieved 2008- 12-21.

13.12 External links

• Understanding big and little endian byte order

• Byte Ordering PPC • Writing endian-independent code in C

This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008 and incorporated under the “relicensing” terms of the GFDL, version 1.3 or later. Chapter 14

Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the “ordinary” (i.e. straight-line) distance between two points in Euclidean space. With this distance, Euclidean space becomes a metric space. The associated norm is called the Euclidean norm. Older literature refers to the metric as Pythagorean metric. A generalized term for the Euclidean norm is the L2 norm or L2 distance.

14.1 Definition

The Euclidean distance between points p and q is the length of the line segment connecting them ( pq ).

In Cartesian coordinates, if p = (p1, p2,..., pn) and q = (q1, q2,..., qn) are two points in Euclidean n-space, then the distance (d) from p to q, or from q to p is given by the Pythagorean formula:

The position of a point in a Euclidean n-space is a Euclidean vector. So, p and q are Euclidean vectors, starting from the origin of the space, and their tips indicate two points. The Euclidean norm, or Euclidean length, or magnitude of a vector measures the length of the vector:

√ √ ∥ ∥ 2 2 ··· 2 · p = p1 + p2 + + pn = p p, where the last equation involves the dot product. A vector can be described as a directed line segment from the origin of the Euclidean space (vector tail), to a point in that space (vector tip). If we consider that its length is actually the distance from its tail to its tip, it becomes clear that the Euclidean norm of a vector is just a special case of Euclidean distance: the Euclidean distance between its tail and its tip. The distance between points p and q may have a direction (e.g. from p to q), so it may be represented by another vector, given by

q − p = (q1 − p1, q2 − p2, ··· , qn − pn)

In a three-dimensional space (n=3), this is an arrow from p to q, which can be also regarded as the position of q relative to p. It may be also called a displacement vector if p and q represent two positions of the same point at two successive instants of time. The Euclidean distance between p and q is just the Euclidean length of this distance (or displacement) vector:

81 82 CHAPTER 14. EUCLIDEAN DISTANCE

which is equivalent to equation 1, and also to:

√ ∥q − p∥ = ∥p∥2 + ∥q∥2 − 2p · q.

14.1.1 One dimension

In one dimension, the distance between two points on the real line is the absolute value of their numerical difference. Thus if x and y are two points on the real line, then the distance between them is given by:

√ (x − y)2 = |x − y|. In one dimension, there is a single homogeneous, translation-invariant metric (in other words, a distance that is induced by a norm), up to a scale factor of length, which is the Euclidean distance. In higher dimensions there are other possible norms.

14.1.2 Two dimensions

In the Euclidean plane, if p = (p1, p2) and q = (q1, q2) then the distance is given by

√ 2 2 d(p, q) = (q1 − p1) + (q2 − p2) . This is equivalent to the Pythagorean theorem.

Alternatively, it follows from (2) that if the polar coordinates of the point p are (r1, θ1) and those of q are (r2, θ2), then the distance between the points is

√ 2 2 − − r1 + r2 2r1r2 cos(θ1 θ2).

14.1.3 Three dimensions

In three-dimensional Euclidean space, the distance is

√ 2 2 2 d(p, q) = (p1 − q1) + (p2 − q2) + (p3 − q3) .

14.1.4 n dimensions

In general, for an n-dimensional space, the distance is

√ 2 2 2 2 d(p, q) = (p1 − q1) + (p2 − q2) + ··· + (pi − qi) + ··· + (pn − qn) .

14.1.5 Squared Euclidean distance

The standard Euclidean distance can be squared in order to place progressively greater weight on objects that are farther apart. In this case, the equation becomes

2 2 2 2 2 d (p, q) = (p1 − q1) + (p2 − q2) + ··· + (pi − qi) + ··· + (pn − qn) . Squared Euclidean Distance is not a metric as it does not satisfy the triangle inequality, however, it is frequently used in optimization problems in which distances only have to be compared. It is also referred to as quadrance within the field of rational trigonometry. 14.2. SEE ALSO 83

14.2 See also

• Chebyshev distance measures distance assuming only the most significant dimension is relevant.

• Euclidean distance matrix • Hamming distance identifies the difference bit by bit of two strings

• Mahalanobis distance normalizes based on a covariance matrix to make the distance metric scale-invariant.

• Manhattan distance measures distance following only axis-aligned directions. • Metric

• Minkowski distance is a generalization that unifies Euclidean distance, Manhattan distance, and Chebyshev distance.

• Pythagorean addition

14.3 References

• Deza, Elena; Deza, Michel Marie (2009). Encyclopedia of Distances. Springer. p. 94.

• “Cluster analysis”. March 2, 2011. Chapter 15

Euclidean space

This article is about Euclidean spaces of all dimensions. For 3-dimensional Euclidean space, see 3-dimensional space. In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria.[1] The term “Euclidean” distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions. Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are also used to define rational numbers. When algebra and mathematical analysis became developed enough, this relation reversed and now it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. It means that points of the space are specified with collections of real numbers, and geometric shapes are defined as equations and inequalities. This approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is essentially only one Euclidean space of each dimension. With Cartesian coor- dinates it is modelled by the real coordinate space (Rn) of the same dimension. In one dimension, this is the real line; in two dimensions, it is the Cartesian plane; and in higher dimensions it is a coordinate space with three or more real number coordinates. Mathematicians denote the n-dimensional Euclidean space by En if they wish to emphasize its Euclidean nature, but Rn is used as well since the latter is assumed to have the standard Euclidean structure, and these two structures are not always distinguished. Euclidean spaces have finite dimension.[2]

15.1 Intuitive overview

One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle. For example, there are two fundamental operations (referred to as motions) on the plane. One is translation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is rotation about a fixed point in the plane, in which every point in the plane turns about that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (usually considered as subsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by some sequence of translations, rotations and reflections (see below). In order to make all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation, and rotation for a mathematically described space. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of units of length and other physical dimensions: the distance in a “mathematical” space is a number, not something expressed in inches or metres. The standard way to define such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product.[2] The reason for working with arbitrary vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner (that is, without choosing a preferred basis). For then:

• the vectors in the vector space correspond to the points of the Euclidean plane,

84 15.1. INTUITIVE OVERVIEW 85

A sphere, the most perfect spatial shape according to Pythagoreans, also is an important concept in modern understanding of Eu- clidean spaces

• the addition operation in the vector space corresponds to translation, and

• the inner product implies notions of angle and distance, which can be used to define rotation.

Once the Euclidean plane has been described in this language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. (However, rotations are more subtle in high dimensions, and visualizing high- dimensional spaces remains difficult, even for experienced mathematicians.) A Euclidean space is not technically a vector space but rather an affine space, on which a vector space acts by transla- tions, or, conversely, a Euclidean vector is the difference (displacement) in an ordered pair of points, not a single point. Intuitively, the distinction says merely that there is no canonical choice of where the origin should go in the space, because it can be translated anywhere. When a certain point is chosen, it can be declared the origin and subsequent calculations may ignore the difference between a point and its coordinate vector, as said above. See point–vector distinction for details. 86 CHAPTER 15. EUCLIDEAN SPACE

Z

(x,y,z)

O z x y Y

X

Every point in three-dimensional Euclidean space is determined by three coordinates.

15.2 Euclidean structure

These are distances between points and the angles between lines or vectors, which satisfy certain conditions (see below), which makes a set of points a Euclidean space. The natural way to obtain these quantities is by introducing and using the standard inner product (also known as the dot product) on Rn.[2] The inner product of any two real n-vectors x and y is defined by

∑n x · y = xiyi = x1y1 + x2y2 + ··· + xnyn, i=1 where xᵢ and yᵢ are ith coordinates of vectors x and y respectively. The result is always a real number.

15.2.1 Distance

Main article: Euclidean distance

The inner product of x with itself is always non-negative. This product allows us to define the “length” of a vector x through square root: 15.2. EUCLIDEAN STRUCTURE 87

v u √ u∑n t 2 ∥x∥ = x · x = (xi) . i=1

This length function satisfies the required properties of a norm and is called the Euclidean norm on Rn. Finally, one can use the norm to define a metric (or distance function) on Rn by

v u u∑n t 2 d(x, y) = ∥x − y∥ = (xi − yi) . i=1

This distance function is called the Euclidean metric. This formula expresses a special case of the Pythagorean theorem. This distance function (which makes a metric space) is sufficient to define all Euclidean geometry, including the dot product. Thus, a real coordinate space together with this Euclidean structure is called Euclidean space. Its vectors form an inner product space (in fact a Hilbert space), and a normed vector space. The metric space structure is the main reason behind the use of real numbers R, not some other ordered field, as the mathematical foundation of Euclidean (and many other) spaces. Euclidean space is a complete metric space, a property which is impossible to achieve operating over rational numbers, for example.

15.2.2 Angle

Main article: Angle The (non-reflex) angle θ (0° ≤ θ ≤ 180°) between vectors x and y is then given by

( ) x · y θ = arccos ∥x∥∥y∥

where arccos is the arccosine function. It is useful only for n > 1,[footnote 1] and the case n = 2 is somewhat special. Namely, on an oriented Euclidean plane one can define an angle between two vectors as a number defined modulo 1 turn (usually denoted as either 2π or 360°), such that ∠y x = −∠x y. This oriented angle is equal either to the angle θ from the formula above or to −θ. If one non-zero vector is fixed (such as the first basis vector), then each non-zero vector is uniquely defined by its magnitude and angle. The angle does not change if vectors x and y are multiplied by positive numbers. Unlike the aforementioned situation with distance, the scale of angles is the same in pure mathematics, physics, and computing. It does not depend on the scale of distances; all distances may be multiplied by some fixed factor, and all angles will be preserved. Usually, the angle is considered a dimensionless quantity, but there are different units of measurement, such as radian (preferred in pure mathematics and theoretical physics) and degree (°) (preferred in most applications).

15.2.3 Rotations and reflections

Main articles: Rotation (mathematics), Reflection (mathematics) and Orthogonal group See also: rotational symmetry and reflection symmetry

Symmetries of a Euclidean space are transformations which preserve the Euclidean metric (called isometries). Al- though aforementioned translations are most obvious of them, they have the same structure for any affine space and do not show a distinctive character of Euclidean geometry. Another family of symmetries leave one point fixed, which may be seen as the origin without loss of generality. All transformations, which preserves the origin and the Euclidean metric, are linear maps. Such transformations Q must, for any x and y, satisfy: 88 CHAPTER 15. EUCLIDEAN SPACE

45°-315°405°

Positive and negative angles on the oriented plane

Qx · Qy = x · y (explain the notation), |Qx| = |x|.

Such transforms constitute a group called the orthogonal group O(n). Its elements Q are exactly solutions of a matrix equation

QTQ = QQT = I, where QT is the transpose of Q and I is the identity matrix. But a Euclidean space is orientable.[footnote 2] Each of these transformations either preserves or reverses orientation depending on whether its determinant is +1 or −1 respectively. Only transformations which preserve orientation, which form the special orthogonal group SO(n), are considered (proper) rotations. This group has, as a Lie group, the same dimension n(n − 1) /2 and is the identity component of O(n). 15.3. NON-CARTESIAN COORDINATES 89

Groups SO(n) are well-studied for n ≤ 4. There are no non-trivial rotations in 0- and 1-spaces. Rotations of a Euclidean plane (n = 2) are parametrized by the angle (modulo 1 turn). Rotations of a 3-space are parametrized with axis and angle, whereas a rotation of a 4-space is a superposition of two 2-dimensional rotations around perpendicular planes. Among linear transforms in O(n) which reverse the orientation are hyperplane reflections. This is the only possible case for n ≤ 2, but starting from three dimensions, such isometry in the general position is a rotoreflection.

15.2.4 Euclidean group

Main article: Euclidean group

The Euclidean group E(n), also referred to as the group of all isometries ISO(n), treats translations, rotations, and reflections in a uniform way, considering them as group actions in the context of group theory, and especially in Lie group theory. These group actions preserve the Euclidean structure. As the group of all isometries, ISO(n), the Euclidean group is important because it makes Euclidean geometry a case of Klein geometry, a theoretical framework including many alternative geometries. The structure of Euclidean spaces – distances, lines, vectors, angles (up to sign), and so on – is invariant under the transformations of their associated Euclidean group. For instance, translations form a commutative subgroup that acts freely and transitively on En, while the stabilizer of any point there is the aforementioned O(n). Along with translations, rotations, reflections, as well as the identity transformation, Euclidean motions comprise also glide reflections (for n ≥ 2), screw operations and rotoreflections (for n ≥ 3), and even more complex combinations of primitive transformations for n ≥ 4. The group structure determines which conditions a metric space needs to satisfy to be a Euclidean space:

1. Firstly, a metric space must be translationally invariant with respect to some (finite-dimensional) real vector space. This means that the space itself is an affine space, that the space is flat, not curved, and points do not have different properties, and so any point can be translated to any other point.

2. Secondly, the metric must correspond in the aforementioned way to some positive-defined quadratic form on this vector space, because point stabilizers have to be isomorphic to O(n).

15.3 Non-Cartesian coordinates

Main article: Coordinate system

Cartesian coordinates are arguably the standard, but not the only possible option for a Euclidean space. Skew coordi- nates are compatible with the affine structure of En, but make formulae for angles and distances more complicated. Another approach, which goes in line with ideas of differential geometry and conformal geometry, is orthogonal coordinates, where coordinate hypersurfaces of different coordinates are orthogonal, although curved. Examples include the polar coordinate system on Euclidean plane, the second important plane coordinate system. See below about expression of the Euclidean structure in curvilinear coordinates.

15.4 Geometric shapes

See also: List of mathematical shapes 90 CHAPTER 15. EUCLIDEAN SPACE

3-dimensional skew coordinates

Parabolic coordinates

15.4.1 Lines, planes, and other subspaces

Main article: Flat (geometry)

The simplest (after points) objects in Euclidean space are flats, or Euclidean subspaces of lesser dimension. Points are 0-dimensional flats, 1-dimensional flats are called (straight) lines, and 2-dimensional flats are planes.(n − 1)- dimensional flats are called hyperplanes. 15.4. GEOMETRIC SHAPES 91

Barycentric coordinates in 3-dimensional space: four coordinates are related with one linear equation

Any two distinct points lie on exactly one line. Any line and a point outside it lie on exactly one plane. More generally, the properties of flats and their incidence of Euclidean space are shared with affine geometry, whereas the affine geometry is devoid of distances and angles.

15.4.2 Line segments and triangles

Main articles: Line segment and Triangle geometry

This is not only a line which a pair (A, B) of distinct points defines. Points on the line which lie between A and B, together with A and B themselves, constitute a line segment AB. Any line segment has the length, which equals to distance between A and B. If A = B, then the segment is degenerate and its length equals to 0, otherwise the length is positive. A (non-degenerate) triangle is defined by three points not lying on the same line. Any triangle lies on one plane. The concept of triangle is not specific to Euclidean spaces, but Euclidean triangles have numerous special properties and define many derived objects. A triangle can be thought of as a 3-gon on a plane, a special (and the first meaningful in Euclidean geometry) case of a polygon. 92 CHAPTER 15. EUCLIDEAN SPACE

Three mutually transversal planes in the 3-dimensional space and their intersections, three lines

15.4.3 Polytopes and root systems

Main articles: Polytope and Root system See also: List of polygons, polyhedra and polytopes and List of regular polytopes

Polytope is a concept that generalizes polygons on a plane and polyhedra in 3-dimensional space (which are among the earliest studied geometrical objects). A simplex is a generalization of a line segment (1-simplex) and a triangle (2-simplex). A tetrahedron is a 3-simplex. The concept of a polytope belongs to affine geometry, which is more general than Euclidean. But Euclidean geometry distinguish regular polytopes. For example, affine geometry does not see the difference between an equilateral triangle and a right triangle, but in Euclidean space the former is regular and the latter is not. Root systems are special sets of Euclidean vectors. A root system is often identical to the set of vertices of a regular polytope.

15.4.4 Curves

Main article: Euclidean geometry of curves See also: List of curves

15.4.5 Balls, spheres, and hypersurfaces

Main articles: Ball (mathematics) and Hypersurface See also: n-sphere and List of surfaces 15.5. TOPOLOGY 93

15.5 Topology

Main article: Real coordinate space § Topological properties

Since Euclidean space is a metric space, it is also a topological space with the natural topology induced by the metric. The metric topology on En is called the Euclidean topology, and it is identical to the standard topology on Rn.A set is open if and only if it contains an open ball around each of its points; in other words, open balls form a base of the topology. The topological dimension of the Euclidean n-space equals n, which implies that spaces of different dimension are not homeomorphic. A finer result is the invariance of domain, which proves that any subset of n-space, that is (with its subspace topology) homeomorphic to an open subset of n-space, is itself open.

15.6 Applications

Aside from countless uses in fundamental mathematics, a Euclidean model of the physical space can be used to solve many practical problems with sufficient precision. Two usual approaches are a fixed, or stationary reference frame (i.e. the description of a motion of objects as their positions that change continuously with time), and the use of Galilean space-time symmetry (such as in Newtonian mechanics). To both of them the modern Euclidean geometry provides a convenient formalism; for example, the space of Galilean velocities is itself a Euclidean space (see relative velocity for details). Topographical maps and technical drawings are planar Euclidean. An idea behind them is the scale invariance of Euclidean geometry, that permits to represent large objects in a small sheet of paper, or a screen.

15.7 Alternatives and generalizations

Although Euclidean spaces are no longer considered to be the only possible setting for a geometry, they act as pro- totypes for other geometric objects. Ideas and terminology from Euclidean geometry (both traditional and analytic) are pervasive in modern mathematics, where other geometric objects share many similarities with Euclidean spaces, share part of their structure, or embed Euclidean spaces.

15.7.1 Curved spaces

Main article: Riemannian geometry

A smooth manifold is a Hausdorff topological space that is locally diffeomorphic to Euclidean space. Diffeomor- phism does not respect distance and angle, but if one additionally prescribes a smoothly varying inner product on the manifold’s tangent spaces, then the result is what is called a Riemannian manifold. Put differently, a Riemannian manifold is a space constructed by deforming and patching together Euclidean spaces. Such a space enjoys notions of distance and angle, but they behave in a curved, non-Euclidean manner. The simplest Riemannian manifold, con- sisting of Rn with a constant inner product, is essentially identical to Euclidean n-space itself. Less trivial examples are n-sphere and hyperbolic spaces. Discovery of the latter in the 19th century was branded as the non-Euclidean geometry. Also, the concept of a Riemannian manifold permits an expression of the Euclidean structure in any smooth coordinate system, via metric tensor. From this tensor one can compute the Riemann curvature tensor. Where the latter equals to zero, the metric structure is locally Euclidean (it means that at least some open set in the coordinate space is isometric to a piece of Euclidean space), no matter whether coordinates are affine or curvilinear.

15.7.2 Indefinite quadratic form

See also: Sylvester’s law of inertia 94 CHAPTER 15. EUCLIDEAN SPACE

If one replaces the inner product of a Euclidean space with an indefinite quadratic form, the result is a pseudo- Euclidean space. Smooth manifolds built from such spaces are called pseudo-Riemannian manifolds. Perhaps their most famous application is the theory of relativity, where flat spacetime is a pseudo-Euclidean space called Minkowski space, where rotations correspond to motions of hyperbolic spaces mentioned above. Further generalization to curved spacetimes form pseudo-Riemannian manifolds, such as in general relativity.

15.7.3 Other number fields

Another line of generalization is to consider other number fields than one of real numbers. Over complex numbers, a Hilbert space can be seen as a generalization of Euclidean dot product structure, although the definition of the inner product becomes a sesquilinear form for compatibility with metric structure.

15.7.4 Infinite dimensions

Main articles: inner product space and Hilbert space

15.8 See also

• Function of several real variables, a coordinate presentation of a function on a Euclidean space • Geometric algebra, an alternative algebraic formalism

• Vector calculus, a standard algebraic formalism

15.9 Footnotes

[1] On the real line (n = 1) any two non-zero vectors are either parallel or antiparallel depending on whether their signs match or oppose. There are no angles between 0 and 180°.

[2] It is Rn which is oriented because of the ordering of elements of the standard basis. Although an orientation is not an attribute of the Euclidean structure, there are only two possible orientations, and any linear automorphism either keeps orientation or reverses (swaps the two).

15.10 References

[1] Ball, W.W. Rouse (1960) [1908]. A Short Account of the History of Mathematics (4th ed.). Dover Publications. pp. 50–62. ISBN 0-486-20630-0.

[2] E.D. Solomentsev (7 February 2011). “Euclidean space.”. Encyclopedia of Mathematics. Springer. Retrieved 1 May 2014.

15.11 External links

• Hazewinkel, Michiel, ed. (2001), “Euclidean space”, Encyclopedia of Mathematics, Springer, ISBN 978-1- 55608-010-4 Chapter 16

Flat (geometry)

In geometry, a flat is a subset of n-dimensional space that is congruent to a Euclidean space of lower dimension. The flats in two-dimensional space are points and lines, and the flats in three-dimensional space are points, lines, and planes. In n-dimensional space, there are flats of every dimension from 0 to n − 1.[1] Flats of dimension n − 1 are called hyperplanes. Flats are similar to linear subspaces, except that they need not pass through the origin. If Euclidean space is considered as an affine space, the flats are precisely the affine subspaces. Flats are important in linear algebra, where they provide a geometric realization of the solution set for a system of linear equations. A flat is also called a linear manifold or linear variety.

16.1 Descriptions

16.1.1 By equations

A flat can be described by a system of linear equations. For example, a line in two-dimensional space can be described by a single linear equation involving x and y:

3x + 5y = 8.

In three-dimensional space, a single linear equation involving x, y, and z defines a plane, while a pair of linear equations can be used to describe a line. In general, a linear equation in n variables describes a hyperplane, and a system of linear equations describes the intersection of those hyperplanes. Assuming the equations are consistent and linearly independent, a system of k equations describes a flat of dimension n − k.

16.1.2 Parametric

A flat can also be described by a system of linear parametric equations. A line can be described by equations involving one parameter:

3 x = 2 + 3t, y = −1 + t z = − 4t 2 while the description of a plane would require two parameters:

x = 5 + 2t1 − 3t2, y = −4 + t1 + 2t2 z = 5t1 − 3t2.

In general, a parameterization of a flat of dimension k would require parameters t1,…, tk.

95 96 CHAPTER 16. FLAT (GEOMETRY)

16.2 Operations and relations on flats

16.2.1 Intersecting, parallel, and skew flats

An intersection of flats is either a flat or the empty set.[2] If every line from the first flat is parallel to some line from the second flat, then these flats are parallel. Two parallel flats of the same dimension either coincide or do not intersect; they can be described by two systems of linear equations which differ only in their right-hand sides. If flats do not intersect, and no line from the first flat is parallel to a line from the second flat, then these are skew flats. It is possible only if sum of their dimensions is less than dimension of the ambient space.

16.2.2 Join

For two flats of dimensions k1 and k2 there exists the minimal flat which contains them, of dimension at most k1 + k2 + 1. If two flats intersect, then the dimension of the containing flat equals to k1 + k2 − dimension of the intersection.

16.2.3 Properties of operations

These two operations (referred to as meet and join) make the set of all flats in the Euclidean n-space a lattice and can build systematic coordinates for flats in any dimension, leading to Grassmann coordinates or dual Grassmann coordinates. For example, a line in three-dimensional space is determined by two distinct points or by two distinct planes.

Though, the lattice of all flats is not a distributive lattice. If two lines ℓ1 and ℓ2 intersect, then ℓ1 ∩ ℓ2 is a point. If p is a point not lying on the same plane, then (ℓ1 ∩ ℓ2) + p = (ℓ1 + p) ∩ (ℓ2 + p), both representing a line. But when ℓ1 and ℓ2 are parallel, this distributivity fails, giving p on the left-hand side and a third parallel line on the right-hand side. The ambient space would be a projective space to accommodate intersections of parallel flats, which lead to objects “at infinity”.

16.3 Euclidean geometry

The aforementioned facts do not depend on the structure being that of Euclidean space (namely, involving Euclidean distance) and are correct in any affine space. In a Euclidean space:

• There is the distance between a flat and a point. (See for example Distance from a point to a plane and Distance from a point to a line.)

• There is the distance between two flats, equal to 0 if they intersect. (See for example Distance between two lines (in the same plane) and Skew lines#Distance.)

• If two flats intersect, then there is the angle between two flats, which belongs to the interval [0, π/2] between 0 and the right angle. (See for example Dihedral angle (between two planes).)

16.4 See also

• N-dimensional space

• Matroid

• Coplanarity 16.5. NOTES 97

16.5 Notes

[1] In addition, all of n-dimensional space is sometimes considered an n-dimensional flat as a subset of itself.

[2] Can be considered as −1-flat.

16.6 References

• Heinrich Guggenheimer (1977) Applicable Geometry,page 7, Krieger, New York.

• Stolfi, Jorge (1991), Oriented Projective Geometry, Academic Press, ISBN 978-0-12-672025-9 From original Stanford Ph.D. dissertation, Primitives for Computational Geometry, available as DEC SRC Research Report 36. • PlanetMath: linear manifold

16.7 External links

• Weisstein, Eric W., “Hyperplane”, MathWorld.

• Weisstein, Eric W., “Flat”, MathWorld. Chapter 17

Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function. For instance, the functions f : X → Y and g : Y → Z can be composed to yield a function which maps x in X to g(f(x)) in Z. Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X.[note 1] The notation g ∘ f is read as "g circle f ", or "g round f ", or "g composed with f ", "g after f ", "g following f ", or "g of f", or "g on f ". Intuitively, composing two functions is a chaining process in which the output of the first function becomes the input of the second function. The composition of functions is a special case of the composition of relations, so all properties of the latter are true of composition of functions.[1] The composition of function has some additional properties.

17.1 Examples

X f Y g Z 1 a @ b 2 # c 3 4 d !!

g ∘ f , the composition of f and g. For example, (g ∘ f )(c) = #.

• Composition of functions on a finite set: If f = {(1,3), (2,1), (3,4), (4,6)}, and g = {(1,5), (2,3), (3,4), (4,1), (5,3), (6,2)}, then g ∘ f = {(1,4), (2,5), (3,1), (4,2)}. • Composition of functions on an infinite set: If f: ℝ → ℝ (where ℝ is the set of all real numbers) is given by f(x) = 2x + 4 and g: ℝ → ℝ is given by g(x) = x3, then:

(f ∘ g)(x) = f(g(x)) = f(x3) = 2x3 + 4, and

98 17.2. PROPERTIES 99

A B C

1 1 1 2 2 2 3 3 3 4 4 4 f : A→B g : B→C

1 1 2 2 3 3 4 4

A g◦f : A→C C

Concrete example for the composition of two function.

(g ∘ f)(x) = g(f(x)) = g(2x + 4) = (2x + 4)3.

• If an airplane’s elevation at time t is given by the function h(t), and the oxygen concentration at elevation x is given by the function c(x), then (c ∘ h)(t) describes the oxygen concentration around the plane at time t.

17.2 Properties

The composition of functions is always associative—a property inherited from the composition of relations.[1] That is, if f, g, and h are three functions with suitably chosen domains and codomains, then f ∘ (g ∘ h) = (f ∘ g) ∘ h, where the parentheses serve to indicate that composition is to be performed first for the parenthesized functions. Since there 100 CHAPTER 17. FUNCTION COMPOSITION is no distinction between the choices of placement of parentheses, they may be left off without causing any ambiguity. In a strict sense, the composition g ∘ f can be built only if f's codomain equals g's domain; in a wider sense it is sufficient that the former is a subset of the latter.[note 2] Moreover, it is often convenient to tacitly restrict f's domain such that f produces only values in g's domain; for example, the composition g ∘ f of the functions f : ℝ → (−∞,+9] defined by f(x) = 9 − x2 and g : [[interval (mathematics)#Infinite endpoints|[0,+∞)]] → ℝ defined by g(x) = √x can be defined on the interval [−3,+3].

y

| f ( x) |

f (| x |) x

f ( x)

Compositions of two real functions, absolute value and a cubic function, in different orders show a non-commutativity of the com- position.

The functions g and f are said to commute with each other if g ∘ f = f ∘ g. Commutativity is a special property, 17.3. COMPOSITION MONOIDS 101

attained only by particular functions, and often in special circumstances. For example, |x| + 3 = |x + 3| only when x ≥ 0. The picture shows another example. The composition of one-to-one functions is always one-to-one. Similarly, the composition of two onto functions is always onto. It follows that composition of two bijections is also a bijection. The inverse function of a composition (assumed invertible) has the property that (f ∘ g)−1 = ( g−1 ∘ f −1).[2] Derivatives of compositions involving differentiable functions can be found using the chain rule. Higher derivatives of such functions are given by Faà di Bruno’s formula.

17.3 Composition monoids

Main article: Transformation monoid

Suppose one has two (or more) functions f: X → X, g: X → X having the same domain and codomain; these are often called transformations. Then one can form chains of transformations composed together, such as f ∘ f ∘ g ∘ f. Such chains have the algebraic structure of a monoid, called a transformation monoid or (much more seldom) composition monoid. In general, transformation monoids can have remarkably complicated structure. One particular notable example is the de Rham curve. The set of all functions f: X → X is called the full transformation semigroup[3] or symmetric semigroup[4] on X. (One can actually define two semigroups depending how one defines the semigroup operation as the left or right composition of functions.[5]) If the transformation are bijective (and thus invertible), then the set of all possible combinations of these functions forms a transformation group; and one says that the group is generated by these functions. A fundamental result in group theory, Cayley’s theorem, essentially says that any group is in fact just a group of permutations (up to isomorphism).[6] The set of all bijective functions f: X → X (called permutations) forms a group with respect to the composition operator. This is the symmetric group, also sometimes called the composition group. In the symmetric semigroup (of all transformations) one also finds a weaker, non-unique notion of inverse (called a pseudoinverse) because the symmetric semigroup is a regular semigroup.[7]

17.4 Functional powers

Main article: Iterated function

If Y ⊆ X, then f: X→Y may compose with itself; this is sometimes denoted as f 2. That is:

(f ∘ f)(x) = f(f(x)) = f 2(x)

(f ∘ f ∘ f)(x) = f(f(f(x))) = f 3(x)

More generally, for any natural number n ≥ 2, the nth functional power can be defined inductively by f n = f ∘ f n−1 = f n−1 ∘ f. Repeated composition of such a function with itself is called iterated function.

• By convention, f 0 is defined as the identity map on f 's domain, idX.

• If even Y = X and f: X → X admits an inverse function f −1, negative functional powers f −n are defined for n > 0 as the opposite power of the inverse function: f −n = (f −1)n.

Note: If f takes its values in a ring (in particular for real or complex-valued f ), there is a risk of confusion, as f n could also stand for the n-fold product of f, e.g. f 2(x) = f(x)· f(x). For trigonometric functions, usually the latter is meant, at least for positive exponents. For example, in trigonometry, this superscript notation represents standard exponentiation when used with trigonometric functions: sin2(x) = sin(x) · sin(x). However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g., tan−1 = arctan (≠ 1/tan). 102 CHAPTER 17. FUNCTION COMPOSITION W T V A U S B C

F 135AS B E

The similarity that transforms triangle EFA into triangle ATB is the composition of a homothety H and a rotation R, of which the common centre is S. For example, the image of A under the rotation R is U, which may be written R (A) = U. And H(U) = B means that the mapping H transforms U into B. Thus H(R (A)) = (H ∘ R )(A) = B.

In some cases, when, for a given function f, the equation g ∘ g = f has a unique solution g, that function can be defined as the functional square root of f, then written as g = f 1/2. More generally, when gn = f has a unique solution for some natural number n > 0, then f m/n can be defined as gm. Under additional restrictions, this idea can be generalized so that the iteration count becomes a continuous parameter; in this case, such a system is called a flow, specified through solutions of Schröder’s equation. Iterated functions and flows occur naturally in the study of fractals and dynamical systems.

17.5 Alternative notations

Many mathematicians, particularly in group theory, omit the composition symbol, writing gf for g ∘ f.[8] In the mid-20th century, some mathematicians decided that writing "g ∘ f " to mean “first apply f, then apply g" was too confusing and decided to change notations. They write "xf " for "f(x)" and "(xf)g" for "g(f(x))".[9] This can be more natural and seem simpler than writing functions on the left in some areas – in linear algebra, for instance, when x is a row vector and f and g denote matrices and the composition is by matrix multiplication. This alternative notation 17.6. COMPOSITION OPERATOR 103

is called postfix notation. The order is important because matrix multiplication is non-commutative. Successive transformations applying and composing to the right agrees with the left-to-right reading sequence. Mathematicians who use postfix notation may write "fg", meaning first apply f and then apply g, in keeping with the order the symbols occur in postfix notation, thus making the notation "fg" ambiguous. Computer scientists may write "f ; g" for this,[10] thereby disambiguating the order of composition. To distinguish the left composition operator from a text semicolon, in the Z notation the character is used for left relation composition.[11] Since all functions are binary relations, it is correct to use the [fat] semicolon for function composition as well (see the article on composition of relations for further details on this notation).

17.6 Composition operator

Main article: Composition operator

Given a function g, the composition operator Cg is defined as that operator which maps functions to functions as

Cgf = f ◦ g.

Composition operators are studied in the field of operator theory.

17.7 In programming languages

Main article: Function composition (computer science)

Function composition appears in one form or another in numerous programming languages.

17.8 Multivariate functions

Partial composition is possible for multivariate functions. The function resulting when some argument xi of the function f is replaced by the function g is called a composition of f and g in some computer engineering contexts, and is denoted f |xi ₌ g

| f xi=g = f(x1, . . . , xi−1, g(x1, x2, . . . , xn), xi+1, . . . , xn). When g is a simple constant b, composition degenerates into a (partial) valuation, whose result is also known as restriction or co-factor.[12]

| f xi=b = f(x1, . . . , xi−1, b, xi+1, . . . , xn). In general, the composition of multivariate functions may involve several other functions as arguments, as in the definition of primitive recursive function. Given f, a n-ary function, and n m-ary functions g1, ..., gn, the composition of f with g1, ..., gn, is the m-ary function

h(x1, . . . , xm) = f(g1(x1, . . . , xm), . . . , gn(x1, . . . , xm))

[13] This is sometimes called the generalized composite of f with g1, ..., gn. The partial composition in only one argu- ment mentioned previously can be instantiated from this more general scheme by setting all argument functions except one to be suitably chosen projection functions. Note also that g1, ..., gn can be seen as a single vector/tuple-valued function in this generalized scheme, in which case this is precisely the standard definition of function composition.[14] 104 CHAPTER 17. FUNCTION COMPOSITION

A set of finitary operations on some base set X is called a clone if it contains all projections and is closed under generalized composition. Note that a clone generally contains operations of various arities.[13] The notion of com- mutation also finds an interesting generalization in the multivariate case; a function f of arity n is said to commute with a function g of arity m if f is a homomorphism preserving g, and vice versa i.e.:[15]

f(g(a11, . . . , a1m), . . . , g(an1, . . . , anm)) = g(f(a11, . . . , an1), . . . , g(a1m, . . . , amn))

A unary operation always commutes with itself, but this is not necessarily the case for a binary (or higher arity) operation. A binary (or higher arity) operation that commutes with itself is called medial or entropic.[15]

17.9 Generalizations

Composition can be generalized to arbitrary binary relations. If R ⊆ X × Y and S ⊆ Y × Z are two binary relations, then their composition S∘R is the relation defined as {(x, z) ∈ X × Z : ∃y ∈ Y.(x, y) ∈ R ∧ (y, z) ∈ S}. Considering a function as a special case of a binary relation (namely functional relations), function composition satisfies the definition for relation composition. The composition is defined in the same way for partial functions and Cayley’s theorem has its analogue called Wagner- Preston theorem.[16] The category of sets with functions as morphisms is the prototypical category. The axioms of a category are in fact inspired from the properties (and also the definition) of function composition.[17] The structures given by compo- sition are axiomatized and generalized in category theory with the concept of morphism as the category-theoretical replacement of functions. The order inversion in the formula (f ∘ g)−1 = (g−1 ∘ f −1) applies for groups in general and for the inverse relation; each of these is a dagger category.

17.10 Typography

The composition symbol ∘ is encoded as U+2218 ∘ ring operator (HTML ∘); see the Degree symbol article for similar-appearing Unicode characters. In TeX, it is written \circ.

17.11 See also

• Combinatory logic

• Function composition (computer science)

• Functional decomposition

• Iterated function

• Flow (mathematics)

• Higher-order function

• Cobweb plot – a graphical technique for functional composition

• Lambda calculus

• Functional square root

• Composition ring, a formal axiomatization of the composition operation

• Function of random variable, distribution of a function of a random variable 17.12. NOTES 105

17.12 Notes

[1] Some authors use f ∘ g : X → Z, defined by (f ∘ g )(x) = g(f(x)) instead.

[2] The strict sense is used, e.g., in category theory, where a subset relation is modelled explicitly by an inclusion function.

17.13 References

[1] Daniel J. Velleman (2006). How to Prove It: A Structured Approach. Cambridge University Press. p. 232. ISBN 978-1- 139-45097-3.

[2] Nancy Rodgers (2000). Learning to Reason: An Introduction to Logic, Sets, and Relations. John Wiley & Sons. pp. 359–362. ISBN 978-0-471-37122-9.

[3] Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 334. ISBN 978-1-4704-1493-1.

[4] Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 2. ISBN 978-0-8247-9662-4.

[5] Pál Dömösi; Chrystopher L. Nehaniv (2005). Algebraic Theory of Automata Networks: A Introduction. SIAM. p. 8. ISBN 978-0-89871-569-9.

[6] Nathan Carter (9 April 2009). Visual Group Theory. MAA. p. 95. ISBN 978-0-88385-757-1.

[7] Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. p. 24. ISBN 978-1-84800-281-4.

[8] Oleg A. Ivanov (1 January 2009). Making Mathematics Come to Life: A Guide for Teachers and Students. American Mathematical Soc. pp. 217–. ISBN 978-0-8218-4808-1.

[9] Jean Gallier (2011). Discrete Mathematics. Springer. p. 118. ISBN 978-1-4419-8047-2.

[10] Michael Barr; Charles Wells (1998). Category Theory for Computing Science (PDF). p. 6. This is the updated and free version of book originally published by Prentice Hall in 1990 as ISBN 978-0-13-120486-7.

[11] ISO/IEC 13568:2002(E), p. 23

[12] Bryant, R.E. (August 1986). “Logic Minimization Algorithms for VLSI Synthesis” (PDF). IEEE Transactions on Computers C–35 (8): 677–691. doi:10.1109/tc.1986.1676819.

[13] Clifford Bergman (2011). Universal Algebra: Fundamentals and Selected Topics. CRC Press. pp. 79–80. ISBN 978-1- 4398-5129-6.

[14] George Tourlakis (2012). Theory of Computation. John Wiley & Sons. p. 100. ISBN 978-1-118-31533-0.

[15] Clifford Bergman (2011). Universal Algebra: Fundamentals and Selected Topics. CRC Press. pp. 90–91. ISBN 978-1- 4398-5129-6.

[16] S. Lipscomb, “Symmetric Inverse Semigroups”, AMS Mathematical Surveys and Monographs (1997), ISBN 0-8218-0627- 0, p. xv

[17] Peter Hilton; Yel-Chiang Wu (1989). A Course in Modern Algebra. John Wiley & Sons. p. 65. ISBN 978-0-471-50405-4.

17.14 External links

• Hazewinkel, Michiel, ed. (2001), “Composite function”, Encyclopedia of Mathematics, Springer, ISBN 978- 1-55608-010-4

• "Composition of Functions" by Bruce Atwood, the Wolfram Demonstrations Project, 2007. Chapter 18

General linear group

For other uses of “GLN”, see GLN.

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the oper- ation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible. The group is so named because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of real numbers) is the group of n×n invertible matrices of real numbers, and is denoted by GLn(R) or GL(n, R). More generally, the general linear group of degree n over any field F (such as the complex numbers), or a ring R (such as the ring of integers), is the set of n×n invertible matrices with entries from F (or R), again with matrix multiplication as the group operation.[1] Typical notation is GLn(F) or GL(n, F), or simply GL(n) if the field is understood. More generally still, the general linear group of a vector space GL(V) is the abstract automorphism group, not nec- essarily written as matrices. The special linear group, written SL(n, F) or SLn(F), is the subgroup of GL(n, F) consisting of matrices with a determinant of 1. The group GL(n, F) and its subgroups are often called linear groups or matrix groups (the abstract group GL(V) is a linear group but not a matrix group). These groups are important in the theory of group representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of polynomials. The modular group may be realised as a quotient of the special linear group SL(2, Z). If n ≥ 2, then the group GL(n, F) is not abelian.

18.1 General linear group of a vector space

If V is a vector space over the field F, the general linear group of V, written GL(V) or Aut(V), is the group of all automorphisms of V, i.e. the set of all bijective linear transformations V → V, together with functional composition as group operation. If V has finite dimension n, then GL(V) and GL(n, F) are isomorphic. The isomorphism is not canonical; it depends on a choice of basis in V. Given a basis (e1, ..., en) of V and an automorphism T in GL(V), we have

∑n T ek = ajkej j=1

for some constants ajk in F; the matrix corresponding to T is then just the matrix with entries given by the ajk. In a similar way, for a commutative ring R the group GL(n, R) may be interpreted as the group of automorphisms of

106 18.2. IN TERMS OF DETERMINANTS 107

a free R-module M of rank n. One can also define GL(M) for any R-module, but in general this is not isomorphic to GL(n, R) (for any n).

18.2 In terms of determinants

Over a field F, a matrix is invertible if and only if its determinant is nonzero. Therefore an alternative definition of GL(n, F) is as the group of matrices with nonzero determinant. Over a commutative ring R, more care is needed: a matrix over R is invertible if and only if its determinant is a unit in R, that is, if its determinant is invertible in R. Therefore GL(n, R) may be defined as the group of matrices whose determinants are units. Over a non-commutative ring R, determinants are not at all well behaved. In this case, GL(n, R) may be defined as the unit group of the matrix ring M(n, R).

18.3 As a Lie group

18.3.1 Real case

The general linear group GL(n, R) over the field of real numbers is a real Lie group of dimension n2. To see this, note that the set of all n×n real matrices, Mn(R), forms a real vector space of dimension n2. The subset GL(n, R) consists of those matrices whose determinant is non-zero. The determinant is a polynomial map, and hence GL(n, R) is an open affine subvariety of Mn(R) (a non-empty open subset of Mn(R) in the Zariski topology), and therefore[2] a smooth manifold of the same dimension.

The Lie algebra of GL(n, R), denoted gln, consists of all n×n real matrices with the commutator serving as the Lie bracket. As a manifold, GL(n, R) is not connected but rather has two connected components: the matrices with positive determinant and the ones with negative determinant. The identity component, denoted by GL+(n, R), consists of the real n×n matrices with positive determinant. This is also a Lie group of dimension n2; it has the same Lie algebra as GL(n, R). The group GL(n, R) is also noncompact. “The”[3] maximal compact subgroup of GL(n, R) is the orthogonal group O(n), while “the” maximal compact subgroup of GL+(n, R) is the special orthogonal group SO(n). As for SO(n), the group GL+(n, R) is not simply connected (except when n = 1), but rather has a fundamental group isomorphic to Z for n = 2 or Z2 for n > 2.

18.3.2 Complex case

The general linear group over the field of complex numbers, GL(n, C), is a complex Lie group of complex dimension n2. As a real Lie group it has dimension 2n2. The set of all real matrices forms a real Lie subgroup. These correspond to the inclusions

GL(n, R) < GL(n, C) < GL(2n, R), which have real dimensions n2, 2n2, and 4n2 = (2n)2. Complex n-dimensional matrices can be characterized as real 2n-dimensional matrices that preserve a linear complex structure — concretely, that commute with a matrix J such that J2 = −I, where J corresponds to multiplying by the imaginary unit i. The Lie algebra corresponding to GL(n, C) consists of all n×n complex matrices with the commutator serving as the Lie bracket. Unlike the real case, GL(n, C) is connected. This follows, in part, since the multiplicative group of complex num- bers C∗ is connected. The group manifold GL(n, C) is not compact; rather its maximal compact subgroup is the unitary group U(n). As for U(n), the group manifold GL(n, C) is not simply connected but has a fundamental group isomorphic to Z. 108 CHAPTER 18. GENERAL LINEAR GROUP

18.4 Over finite fields

Cayley table of GL(2, 2), which is isomorphic to S3.

If F is a finite field with q elements, then we sometimes write GL(n, q) instead of GL(n, F). When p is prime, GL(n, p) is the outer automorphism group of the group Zpn, and also the automorphism group, because Zpn is abelian, so the inner automorphism group is trivial. The order of GL(n, q) is:

(qn − 1)(qn − q)(qn − q2) ··· (qn − qn−1)

This can be shown by counting the possible columns of the matrix: the first column can be anything but the zero vector; the second column can be anything but the multiples of the first column; and in general, the kth column can n (n) be any vector not in the linear span of the first k − 1 columns. In q-analog notation, this is [n]q!(q − 1) q 2 . For example, GL(3, 2) has order (8 − 1)(8 − 2)(8 − 4) = 168. It is the automorphism group of the Fano plane and of 3 the group Z2 , and is also known as PSL(2, 7). More generally, one can count points of Grassmannian over F: in other words the number of subspaces of a given 18.5. SPECIAL LINEAR GROUP 109

dimension k. This requires only finding the order of the stabilizer subgroup of one such subspace and dividing into the formula just given, by the orbit-stabilizer theorem. These formulas are connected to the Schubert decomposition of the Grassmannian, and are q-analogs of the Betti numbers of complex Grassmannians. This was one of the clues leading to the Weil conjectures. Note that in the limit q ↦ 1 the order of GL(n, q) goes to 0! – but under the correct procedure (dividing by (q − 1)n) we see that it is the order of the symmetric group (See Lorscheid’s article) – in the philosophy of the field with one element, one thus interprets the symmetric group as the general linear group over the field with one element: S ≅ GL(n, 1).

18.4.1 History

The general linear group over a prime field, GL(ν, p), was constructed and its order computed by Évariste Galois in 1832, in his last letter (to Chevalier) and second (of three) attached manuscripts, which he used in the context of studying the Galois group of the general equation of order pν.[4]

18.5 Special linear group

Main article: Special linear group

The special linear group, SL(n, F), is the group of all matrices with determinant 1. They are special in that they lie on a subvariety – they satisfy a polynomial equation (as the determinant is a polynomial in the entries). Matrices of this type form a group as the determinant of the product of two matrices is the product of the determinants of each matrix. SL(n, F) is a normal subgroup of GL(n, F). If we write F× for the multiplicative group of F (excluding 0), then the determinant is a group homomorphism

det: GL(n, F) → F×.

that is surjective and its kernel is the special linear group. Therefore, by the first isomorphism theorem, GL(n, F)/SL(n, F) is isomorphic to F×. In fact, GL(n, F) can be written as a semidirect product:

GL(n, F) = SL(n, F) ⋊ F×

When F is R or C, SL(n, F) is a Lie subgroup of GL(n, F) of dimension n2 − 1. The Lie algebra of SL(n, F) consists of all n×n matrices over F with vanishing trace. The Lie bracket is given by the commutator. The special linear group SL(n, R) can be characterized as the group of volume and orientation preserving linear transformations of Rn. The group SL(n, C) is simply connected, while SL(n, R) is not. SL(n, R) has the same fundamental group as GL+(n, R), that is, Z for n = 2 and Z2 for n > 2.

18.6 Other subgroups

18.6.1 Diagonal subgroups

The set of all invertible diagonal matrices forms a subgroup of GL(n, F) isomorphic to (F×)n. In fields like R and C, these correspond to rescaling the space; the so-called dilations and contractions. A scalar matrix is a diagonal matrix which is a constant times the identity matrix. The set of all nonzero scalar matrices forms a subgroup of GL(n, F) isomorphic to F× . This group is the center of GL(n, F). In particular, it is a normal, abelian subgroup. The center of SL(n, F) is simply the set of all scalar matrices with unit determinant, and is isomorphic to the group of nth roots of unity in the field F. 110 CHAPTER 18. GENERAL LINEAR GROUP

18.6.2 Classical groups

The so-called classical groups are subgroups of GL(V) which preserve some sort of bilinear form on a vector space V. These include the

• orthogonal group, O(V), which preserves a non-degenerate quadratic form on V, • symplectic group, Sp(V), which preserves a symplectic form on V (a non-degenerate alternating form), • unitary group, U(V), which, when F = C, preserves a non-degenerate hermitian form on V.

These groups provide important examples of Lie groups.

18.7 Related groups and monoids

18.7.1 Projective linear group

Main article: Projective linear group

The projective linear group PGL(n, F) and the projective special linear group PSL(n, F) are the quotients of GL(n, F) and SL(n, F) by their centers (which consist of the multiples of the identity matrix therein); they are the induced action on the associated projective space.

18.7.2 Affine group

Main article: Affine group

The affine group Aff(n, F) is an extension of GL(n, F) by the group of translations in Fn. It can be written as a semidirect product:

Aff(n, F) = GL(n, F) ⋉ Fn

where GL(n, F) acts on Fn in the natural manner. The affine group can be viewed as the group of all affine transfor- mations of the affine space underlying the vector space Fn. One has analogous constructions for other subgroups of the general linear group: for instance, the special affine group is the subgroup defined by the semidirect product, SL(n, F) ⋉ Fn, and the Poincaré group is the affine group associated to the Lorentz group, O(1, 3, F) ⋉ Fn.

18.7.3 General semilinear group

Main article: General semilinear group

The general semilinear group ΓL(n, F) is the group of all invertible semilinear transformations, and contains GL. A semilinear transformation is a transformation which is linear “up to a twist”, meaning “up to a field automorphism under scalar multiplication”. It can be written as a semidirect product:

ΓL(n, F) = Gal(F) ⋉ GL(n, F) where Gal(F) is the Galois group of F (over its prime field), which acts on GL(n, F) by the Galois action on the entries. The main interest of ΓL(n, F) is that the associated projective semilinear group PΓL(n, F) (which contains PGL(n, F)) is the collineation group of projective space, for n > 2, and thus semilinear maps are of interest in projective geometry. 18.8. INFINITE GENERAL LINEAR GROUP 111

18.7.4 Full linear monoid

If one removes the restriction of the determinant being non-zero, the resulting algebraic structure is a monoid, usually called the full linear monoid,[5][6][7] but occasionally also full linear semigroup,[8] general linear monoid[9][10] etc. It is actually a regular semigroup.[6]

18.8 Infinite general linear group

The infinite general linear group or stable general linear group is the direct limit of the inclusions GL(n, F) → GL(n + 1, F) as the upper left block matrix. It is denoted by either GL(F) or GL(∞, F), and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places.[11]

It is used in algebraic K-theory to define K1, and over the reals has a well-understood topology, thanks to Bott periodicity. It should not be confused with the space of (bounded) invertible operators on a Hilbert space, which is a larger group, and topologically much simpler, namely contractible – see Kuiper’s theorem.

18.9 See also

• List of finite simple groups

• SL2(R)

• Representation theory of SL2(R)

18.10 Notes

[1] Here rings are assumed to be associative and unital.

[2] Since the Zariski topology is coarser than the metric topology; equivalently, polynomial maps are continuous.

[3] A maximal compact subgroup is not unique, but is essentially unique, hence one often refers to “the” maximal compact subgroup.

[4] Galois, Évariste (1846). “Lettre de Galois à M. Auguste Chevalier”. Journal de Mathématiques Pures et Appliquées XI: 408–415. Retrieved 2009-02-04, GL(ν,p) discussed on p. 410.

[5] Jan Okniński (1998). Semigroups of Matrices. World Scientific. Chapter 2: Full linear monoid. ISBN 978-981-02-3445-4.

[6] Meakin (2007). “Groups and Semigroups: Connections and contrast”. In C. M. Campbell. Groups St Andrews 2005. Cambridge University Press. p. 471. ISBN 978-0-521-69470-4.

[7] John Rhodes; Benjamin Steinberg (2009). The q-theory of Finite Semigroups. Springer Science & Business Media. p. 306. ISBN 978-0-387-09781-7.

[8] Eric Jespers; Jan Okniski (2007). Noetherian Semigroup Algebras. Springer Science & Business Media. 2.3: Full linear semigroup. ISBN 978-1-4020-5810-3.

[9] Meinolf Geck (2013). An Introduction to Algebraic Geometry and Algebraic Groups. Oxford University Press. p. 132. ISBN 978-0-19-967616-3.

[10] Mahir Bilen Can; Zhenheng Li; Benjamin Steinberg; Qiang Wang (2014). Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics. Springer. p. 142. ISBN 978-1-4939-0938-4.

[11] Milnor, John Willard (1971). Introduction to algebraic K-theory. Annals of Mathematics Studies 72. Princeton, NJ: Princeton University Press. p. 25. MR 0349811. Zbl 0237.18005. 112 CHAPTER 18. GENERAL LINEAR GROUP

18.11 External links

• Hazewinkel, Michiel, ed. (2001), “General linear group”, Encyclopedia of Mathematics, Springer, ISBN 978- 1-55608-010-4 • “GL(2, p) and GL(3, 3) Acting on Points” by Ed Pegg, Jr., Wolfram Demonstrations Project, 2007. Chapter 19

Glide reflection

Example of a glide reflection: A composite of a reflection across a line and a translation parallel to the line of reflection

A glide reflection will map a set of left and right footprints into each other

In 2-dimensional geometry, a glide reflection (or transflection) is a type of opposite isometry of the Euclidean plane: the combination of a reflection in a line and a translation along that line. A single glide is represented as frieze group p11g. A glide reflection can be seen as a limiting rotoreflection, where the rotation becomes a translation. It can also be given a Schoenflies notation as S₂∞, Coxeter notation as [∞+,2+], and orbifold notation as ∞×.

19.1 Description

The combination of a reflection in a line and a translation in a perpendicular direction is a reflection in a parallel line. However, a glide reflection cannot be reduced like that. Thus the effect of a reflection combined with any translation is a glide reflection, with as special case just a reflection. These are the two kinds of indirect isometries in 2D. For example, there is an isometry consisting of the reflection on the x-axis, followed by translation of one unit parallel to it. In coordinates, it takes

113 114 CHAPTER 19. GLIDE REFLECTION

(x, y) → (x + 1, −y).

It fixes a system of parallel lines. The isometry group generated by just a glide reflection is an infinite cyclic group.[1] Combining two equal glide reflections gives a pure translation with a translation vector that is twice that of the glide reflection, so the even powers of the glide reflection form a translation group. In the case of glide reflection symmetry, the symmetry group of an object contains a glide reflection, and hence the group generated by it. If that is all it contains, this type is frieze group p11g. Example pattern with this symmetry group:

Frieze group nr. 6 (glide-reflections, translations and rotations) is generated by a glide reflection and a rotation about a point on the line of reflection. It is isomorphic to a semi-direct product of Z and C2. Example pattern with this symmetry group:

A typical example of glide reflection in everyday life would be the track of footprints left in the sand by a person walking on a beach. For any symmetry group containing some glide reflection symmetry, the translation vector of any glide reflection is one half of an element of the translation group. If the translation vector of a glide reflection is itself an element of the translation group, then the corresponding glide reflection symmetry reduces to a combination of reflection symmetry and translational symmetry. Glide reflection symmetry with respect to two parallel lines with the same translation implies that there is also trans- lational symmetry in the direction perpendicular to these lines, with a translation distance which is twice the distance between glide reflection lines. This corresponds to wallpaper group pg; with additional symmetry it occurs also in pmg, pgg and p4g. If there are also true reflection lines in the same direction then they are evenly spaced between the glide reflection lines. A glide reflection line parallel to a true reflection line already implies this situation. This corresponds to wallpaper group cm. The translational symmetry is given by oblique translation vectors from one point on a true reflection line to two points on the next, supporting a rhombus with the true reflection line as one of the diagonals. With additional symmetry it occurs also in cmm, p3m1, p31m, p4m and p6m. In 3D the glide reflection is called a glide plane. It is a reflection in a plane combined with a translation parallel to the plane.

19.2 Wallpaper groups

In the Euclidean plane 3 of 17 wallpaper groups require glide reflection generators. p2gg has orthogonal glide reflec- tions and 2-fold rotations. cm has parallel mirrors and glides, and pg has parallel glides. (Glide reflections are shown below as dashed lines)

19.3 Glide reflection in nature and games

Glide symmetry can be observed in nature among certain fossils of the Ediacara biota; the machaeridians; and certain palaeoscolecid worms.[2] Glide reflection is common in Conway’s Game of Life. 19.4. SEE ALSO 115

19.4 See also

• Screw axis, glide plane for the corresponding 3D symmetry operations

19.5 References

[1] Martin, George E. (1982), Transformation Geometry: An Introduction to Symmetry, Undergraduate Texts in Mathematics, Springer, p. 64, ISBN 9780387906362.

[2] Waggoner, B. M. (1996). “Phylogenetic Hypotheses of the Relationships of Arthropods to Precambrian and Cambrian Problematic Fossil Taxa”. Systematic Biology 45 (2): 190–222. doi:10.2307/2413615. JSTOR 2413615.

19.6 External links

• Glide Reflection at cut-the-knot Chapter 20

Homography

This article is about the mathematical notion. For other uses, see Homography (disambiguation).

In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which they are derived.[1] It is a bijection that maps lines to lines, and thus a collineation. In general, there are collineations which are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation. Historically, homographies (and projective spaces) have been introduced to study perspective and projections in Euclidean geometry, and the term “homography”, which, etymologically, roughly means “similar drawing” date from this time. At the end of 19th century, formal definitions of projective spaces were introduced, which differed from extending Euclidean or affine spaces by adding points at infinity. The term “projective transformation” originated in these abstract constructions. These constructions divide into two classes that have been shown to be equivalent. A projective space may be constructed as the set of the lines of a vector space over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see also synthetic geome- try); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called “projective collineations”. For sake of simplicity, unless otherwise stated, the projective spaces considered in this article are supposed to be defined over a (commutative) field. Equivalently Pappus’s hexagon theorem and Desargues’ theorem are supposed to be true. A large part of the results remain true, or may be generalized to projective geometries for which these theorems do not hold.

20.1 Geometric motivation

Historically, the concept of homography had been introduced to understand, explain and study visual perspective, and, specifically, the difference in appearance of two plane objects viewed from different points of view. In the Euclidean space of dimension 3, a central projection from a point O (the center) onto a plane P which does not contain O is the mapping sending a point A to the intersection (if it exists) of the line OA and the plane P. The projection is not defined if the point A belongs to the plane passing through O and parallel to P. The notion of projective space was originally introduced by extending the Euclidean space, that is, by adding points at infinity to it, in order to define the projection for every point except O. Given another plane Q, which does not contain O, the restriction to Q of the above projection is called a perspectivity. With these definitions, a perspectivity is only a partial function, but it becomes a bijection if extended to projective spaces. Therefore, this notion is normally defined for projective spaces. The notion is also easily generalized to projective spaces of any dimension, over any field, in the following way: given two projective spaces P and Q of dimension n, a perspectivity is a bijection from P to Q, which may be obtained by embedding P and Q in a projective space R of dimension n+1 and restricting to P a central projection onto Q.

116 20.2. DEFINITION AND EXPRESSION IN HOMOGENEOUS COORDINATES 117

A' C' B' D'

A B D C

Points A, B, C, D and A', B', C', D' are related by a perspectivity, which is a projective transformation.

If f is a perspectivity from P to Q, and g a perspectivity from Q to P, with a different center, then g∘f is a homography from P to itself, which is called a central collineation, when the dimension of P is at least two. (see below and Perspectivity#Perspective collineations). Originally, a homography was defined as the composition of a finite number of perspectivities.[2] It is a part of the fundamental theorem of projective geometry (see below) that this definition coincides with the more algebraic definition sketched in the introduction and detailed below.

20.2 Definition and expression in homogeneous coordinates

A projective space P(V) of dimension n over a field K may be defined as the set of the lines in a K-vector space of n+1 dimension n+1. If a basis of V has been fixed, a point of V may be represented by a point (x0, . . . , xn) of K .A point of P(V), being a line in V, may thus be represented by the coordinates of any nonzero point of this line, which are thus called homogeneous coordinates of the projective point. Given two projective spaces P(V) and P(W) of the same dimension, an homography is a mapping from P(V) to P(W), which is induced by an isomorphism of vector spaces f : V → W . Such an isomorphism induces a bijection from P(V) to P(W), because of the linearity of f. Two such isomorphisms, f and g, define the same homography if 118 CHAPTER 20. HOMOGRAPHY

and only if there is a nonzero element a of K such that g = af. This may be written in terms of homogeneous coordinates in the following way: A homography φ may be defined by a nonsingular n+1 × n+1 matrix [ai,j], called the matrix of the homography. This matrix is defined up to the multiplication by a nonzero element of K. The homogeneous coordinates [x0 : ··· : xn] of a point and the coordinates [y0 : ··· : yn] of its image by φ are related by

y0 = a0,0x0 + ··· + a0,nxn . .

yn = an,0x0 + ··· + an,nxn. When the projective spaces are defined by adding points at infinity to affine spaces (projective completion) the pre- ceding formulas become, in affine coordinates,

a1,0 + a1,1x1 + ··· + a1,nxn y1 = a0,0 + a0,1x1 + ··· + a0,nxn . .

an,0 + an,1x1 + ··· + an,nxn yn = a0,0 + a0,1x1 + ··· + a0,nxn which generalizes the expression of the homographic function of the next section. This defines only a partial function between affine spaces, which is defined only outside the hyperplane where the denominator is zero.

20.3 Homographies of a projective line

The projective line over a field K may be identified with the union of K and a point, called the “point at infinity” and denoted by ∞ (see projective line). With this representation of the projective line, the homographies are the mappings

az + b z 7→ , where ad − bc ≠ 0, cz + d which are called homographic functions or linear fractional transformations. In the case of the complex projective line, also called the Riemann sphere, the homographies are called Möbius transformations. These correspond precisely with those bijections of the Riemann sphere which preserve orientation and are conformal.[3] In the study of collineations, the case of projective lines is special due to the small dimension. When the line is viewed as a projective space in isolation, any permutation of the points of a projective line is a collineation,[4] since every set of points are collinear. However, if the projective line is embedded in a higher-dimensional projective space, the geometric structure of that space can be used to impose a geometric structure on the line. Thus, in synthetic geometry, the homographies and the collineations of the projective line that are considered are those which are obtained by restrictions to the line of collineations and homographies of spaces of higher dimension. This means that the fundamental theorem of projective geometry (see below) remains valid in the one-dimensional setting. A homography of a projective line may also be properly defined by insisting that the mapping preserves cross-ratios.[5]

20.4 Projective frame and coordinates

A projective frame or projective basis of a projective space of dimension n is an ordered set of n+2 points such no hyperplane contains n+1 of them. A projective frame is sometimes called a simplex,[6] although a simplex in a space of dimension n has at most n+1 vertices. In this section, we consider projective spaces over a commutative field K, although most results may be generalized to projective spaces over a division algebra. Thus, we consider a projective space P(V) of dimension n, where V is 20.4. PROJECTIVE FRAME AND COORDINATES 119

Homographies of the complex plane preserve orthogonal circles

a K-vector space of dimension n+1. Let p : V \{0} → P (V ) be the canonical projection which maps a nonzero vector to the vector line that contains it.

Given a frame (p(e0), . . . , p(en+1)) of P(V), the definition implies the existence of nonzero elements of K such that λ0e0 + ··· + λn+1en+1 = 0 . Replacing ei by λiei for i ≤ n and en+1 by −λn+1en+1 , we get the following characterization of a frame: n+2 points of P(V) form a frame if and only if they are the image by p of a basis of V and the sum of its elements. Moreover, two bases define the same frame in this way, if and only if the elements of the second one are the products of the elements of the first one by a fixed nonzero element of K. It follows that, given two frames, there is exactly one homography mapping the first one onto the second one. In particular, the only homography fixing the points of a frame is the identity map. This result is much more difficult in synthetic geometry (where projective spaces are defined through axioms). It is sometimes called the first fundamental theorem of projective geometry.[7]

Every frame (p(e0), . . . , p(en), p(e0 +···+en)) allows to define projective coordinates, also known as homogeneous coordinates: every point may be written as p(v); the projective coordinates of p(v) on this frame are the coordinates of v on the base (e0, . . . , en). It is not difficult to verify that changing the ei and v, without changing the frame nor p(v), results in multiplying the projective coordinates by the same nonzero element of K. One may also consider the projective space P(Kn+1). It has a canonical frame consisting of the image by p of the canonical basis of Kn+1 (consisting of the elements having only one non zero entry, which is equal to 1), and (1, 1, ..., 1). On this basis, the homogeneous coordinates of p(v) are simply the entries (coefficients) of v. Given another projective space P(V) of the same dimension, and a frame F of it, there is one homography h mapping F onto the canonical frame of P(Kn+1). The projective coordinates of a point a on the frame F are the homogeneous coordinates of h(a) on the canonical frame of P(Kn+1). 120 CHAPTER 20. HOMOGRAPHY

20.5 Central collineations

A' C' B' D'

A B D C

Points A, B, C, D and A', B', C', D' are related by several central collineations, which are completely specified by choosing a line of fixed points L passing through the intersection of the lines ABCD and A'B'C'D'. Let O the intersection of the lines AA', BB', CC', DD'. The image E' of a point E by this collineation is the intersection of the lines A'I and OE, where I is the intersection of the lines L and AE.

In above sections, homographies have been defined through linear algebra. In synthetic geometry, they are traditionally defined as the composition of one or several special homographies called central collineations. It is a part of the fundamental theorem of projective geometry that the two definitions are equivalent. In a projective space, P, of dimension n ≥ 2, a collineation of P is a bijection from P onto P that maps lines onto lines. A central collineation (traditionally these were called perspectivities,[8] but this term may be confusing, having another meaning; see Perspectivity) is a bijection α from P to P, such that there exists a hyperplane H (called the axis of α) which is fixed pointwise by α (that is, α(X) = X for all points X in H) and a point O (called the center of α) which is fixed linewise by α (any line through O is mapped to itself by α, but not necessarily pointwise).[9] There are two types of central collineations. Elations are the central collineations in which the center is incident with the axis and homologies are those in which the center is not incident with the axis. Not that a central collineation is uniquely defined by its center, its axis, a point and its image. A central collineation is a homography that may be defined by a (n+1) × (n+1) matrix which has an eigenspace of dimension n. It is a homology, if the matrix has another eigenvalue and is therefore diagonalizable. It is an elation, if 20.6. FUNDAMENTAL THEOREM OF PROJECTIVE GEOMETRY 121

all the eigenvalues are equal and the matrix is not diagonalizable. The geometric view of a central collineation is easiest to see in a projective plane. Given a central collineation α, consider a line ℓ which does not pass through the center O, and its image under α, ℓ′ = α(ℓ) . Setting R = ℓ ∩ ℓ′ , the axis of α is some line M through R. The image of any point A of ℓ under α is the intersection of OA with ℓ′ . The image B' of a point B that does not belong to ℓ may be constructed in the following way: let S = AB ∩ M, then B′ = SA′ ∩ OB. The composition of two central collineations, while still a homography, will in general not be a central collineation. In fact every homography is the composition of a finite number of central collineations. In synthetic geometry, this prop- erty, which is a part of the fundamental theory of projective geometry is taken as the definition of homographies.[10]

20.6 Fundamental theorem of projective geometry

See also: Collineation § Fundamental theorem of projective geometry and Perspectivity

There are collineations besides the homographies. In particular, any field automorphism σ of a field F induces a collineation of every projective space over F by applying σ to all homogeneous coordinates (over a projective frame) of a point. These collineations are called automorphic collineations. The fundamental theorem of projective geometry consists of the three following theorems.

1. Given two projective frames of a projective space P, there is exactly one homography of P that maps the first frame onto the second one.

2. If the dimension of a projective space P is at least two, every collineation of P is the composition of an auto- morphic collineation and a homography. In particular, over the reals, every collineation of a projective space of dimension at least two is a homography.[11]

3. Every homography is the composition of a finite number of perspectivities. In particular, if the dimension of the implied projective space is at least two, every homography is the composition of a finite number of central collineations.

If projective spaces are defined by means of axioms (synthetic geometry), the third part is simply a definition. On the other hand, if projective spaces are defined by means of linear algebra, the first part is an easy corollary of the definitions. Therefore, the proof of the first part in synthetic geometry, and the proof of the third part in terms of linear algebra both are fundamental steps of the proof of the equivalence of the two ways of defining projective spaces.

20.7 Homography groups

As every homography has an inverse mapping and the composition of two homographies is another, the homographies of a given projective space form a group. For example, the Möbius group is the homography group of any complex projective line. As all the projective spaces of the same dimension over the same field are isomorphic, the same is true for their homography groups. They are therefore considered as a single group acting on several spaces, and only the dimension and the field appear in the notation, not the specific projective space. Homography groups also called projective linear groups are denoted PGL(n + 1, F) when acting on a projective space of dimension n over a field F. Above definition of homographies shows that PGL(n + 1, F) may be identified to the quotient group GL(n + 1, F)/ F∗I, where GL(n + 1, F) is the general linear group of the invertible matrices, and F∗I is the group of the products by a nonzero element of F of the identity matrix of size (n + 1) × (n + 1). When F is a Galois field GF(q) then the homography group is written PGL(n,q). For example, PGL(2,7) acts on the eight points in the projective line over the finite field GF(7), while PGL(2,4), which is isomorphic to the alternating [12] group A5, is the homography group of the projective line with five points. The homography group PGL(n + 1, F) is a subgroup of the collineation group PΓL(n + 1,F) of the collineations of a projective space of dimension n. When the points and lines of the projective space are viewed as a block design, 122 CHAPTER 20. HOMOGRAPHY whose blocks are the sets of points contained in a line, it is common to call the collineation group the automorphism group of the design.

20.8 Cross-ratio

Main article: Cross-ratio

The cross-ratio of four collinear points is an invariant under the homography which is fundamental for the study of the homographies of the lines. Three distinct points a, b and c on a projective line over a field F form a projective frame of this line. There is therefore a unique homography h of this line onto F ∪ ∞ that maps a to ∞, b to 0, and c to 1. Given a fourth point on the same line, the cross-ratio of the four points a, b, c and d, denoted [a, b; c, d], is the element h(d) of F ∪ ∞. In other words, if d has homogeneous coordinates [k : 1] over the projective frame (a, b, c), then [a, b; c, d] = k.[13]

20.9 Over a ring

Main article: projective line over a ring

Suppose A is a ring and U is its group of units. Homographies act on a projective line over A, written P(A), consisting of points U(a, b) with homogeneous coordinates. The homographies on P(A) are described by matrix mappings

( ) a c U(z, 1) = U(za + b, zc + d). b d When A is a commutative ring, the homography may be written

za + b z 7→ , zc + d but otherwise the linear fractional transformation is seen as an equivalence:

U(za + b, zc + d) ∼ U((zc + d)−1(za + b), 1). Ring homographies have been used in quaternion analysis, and with dual quaternions to facilitate screw theory. When A is taken to be biquaternions the homographies exhibit conformal symmetry of an electromagnetic field. The ho- mography group of the ring of integers Z is modular group PSL(2,Z).

20.10 Periodic homographies ( ) ( ) ( ) 1 1 1 n 1 0 The homography h = is periodic when the ring is Z (mod n) since then hn = = . Arthur 0 1 0 1 0 1 Cayley was interested in periodicity when he calculated iterates in 1879.[14] In his review of a brute force approach to periodicity of homographies, H.S.M. Coxeter gave this analysis:

A real homography is involutory (of period 2) if and only if a + d = 0. If it is periodic with period n > 2, then it is elliptic, and no loss of generality occurs by assuming that ad − bc = 1. Since the characteristic roots are exp(± hπi/m), where (h, m) = 1, the trace is a + d = 2 cos(hπ/m).[15]

20.11 See also

• W-curve 20.12. NOTES 123

20.12 Notes

[1] Berger, chapter 4

[2] Meserve 1983, pp. 43-4

[3] Hartshorne 1967, p. 138

[4] Yale 1968, p. 244, Baer 2005, p. 50, Artin 1957, p. 88

[5] In older treatments one often sees the requirement of preserving harmonic tetrads (harmonic sets) (four collinear points whose cross-ratio is −1) but this excludes projective lines defined over fields of characteristic two and so is unnecessarily restrictive. See Baer 2005, p. 76

[6] Baer, p. 66

[7] Berger, chapter 6

[8] Yale 1968, p. 224

[9] Beutelspacher & Rosenbaum 1998, p. 96

[10] Meserve 1983, pp. 43-4

[11] Hirschfeld 1979, p. 30

[12] Hirschfeld 1979, p. 129

[13] Berger, chapter 6   a b [14] Arthur Cayley (1879) “On the matrix  , and its connection with the function ax+b ", Messenger of Mathematics c d cx+d 9:104

[15] H.S.M. Coxeter, On periodicity in Mathematical Reviews

20.13 References

• Artin, E. (1957), Geometric Algebra, Interscience Publishers • Baer, Reinhold (2005) [First published 1952], Linear Algebra and Projective Geometry, Dover, ISBN 9780486445656 • Berger, Marcel (2009), Geometry I, Springer-Verlag, ISBN 978-3-540-11658-5, translated from the 1977 French original by M. Cole and S. Levy, fourth printing of the 1987 English translation • Beutelspacher, Albrecht; Rosenbaum, Ute (1998), Projective Geometry: From Foundations to Applications, Cambridge University Press, ISBN 0-521-48364-6 • Hartshorne, Robin (1967), Foundations of Projective Geometry, New York: W.A. Benjamin, Inc • Hirschfeld, J. W. P. (1979), Projective Geometries Over Finite Fields, Oxford University Press, ISBN 978-0- 19-850295-1 • Meserve, Bruce E. (1983), Fundamental Concepts of Geometry, Dover, ISBN 0-486-63415-9 • Yale, Paul B. (1968), Geometry and Symmetry, Holden-Day

20.14 Further reading

• Patrick du Val (1964) Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, MR 0169108 . • Gunter Ewald (1971) Geometry: An Introduction, page 263, Belmont:Wadsworth Publishing ISBN 0-534- 00034-7. Chapter 21

Homothetic transformation

D E E D A

C A C S

B B

Two similar geometric figures related by a homothetic transformation with respect to a homothetic center S. The angles at corre- sponding points are the same and have the same sense; for example, the angles ABC and A'B'C' are both clockwise and equal in magnitude.

In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends

−−→ M 7→ S + λSM, in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ.[1] In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L. In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the line at infinity pointwise invariant.[2] In Euclidean geometry, a homothety of ratio λ multiplies distances between points by |λ| and all areas by λ2. The first number is called the ratio of magnification or dilation factor or scale factor or similitude ratio. Such a transformation can be called an enlargement if the scale factor exceeds 1. The above-mentioned fixed point S is called homothetic center or center of similarity or center of similitude

21.1 Homothety and uniform scaling

If the homothetic center S happens to coincide with the origin O of the vector space (S ≡ O), then every homothety with scale factor λ is equivalent to a uniform scaling by the same factor, which sends

−−→ −−→ OM 7→ λOM.

124 21.2. SEE ALSO 125

As a consequence, in the specific case in which S ≡ O, the homothety becomes a linear transformation, which preserves not only the collinearity of points (straight lines are mapped to straight lines), but also vector addition and scalar multiplication. The image of a point (x, y) after a homothety with center (a, b) and scale factor λ is given by (a + λ(x − a), b + λ(y − b)).

21.2 See also

• Scaling (geometry) a similar notion in vector spaces

• Homothetic center, the center of a homothetic transformation taking one of a pair of shapes into the other • The Hadwiger conjecture on the number of strictly smaller homothetic copies of a convex body that may be needed to cover it • Homothetic function (economics), a function of the form f(U(y)) in which U is a homogeneous function and f is a monotonically increasing function.

21.3 Notes

[1] Hadamard (, p. 145)

[2] Tuller (, p. 119)

21.4 References

• Hadamard, J., Lessons in Plane Geometry.

• Meserve, Bruce E. (1955), “Homothetic transformations”, Fundamental Concepts of Geometry, Addison- Wesley, pp. 166–169.

• Tuller, Annita, A Modern Introduction to Geometries.

21.5 External links

• Homothety, interactive applet from Cut-the-Knot. Chapter 22

Improper rotation

A pentagonal antiprism with marked edges shows rotoreflectional symmetry, with an order of 10.

In geometry, an improper rotation,[1] also called rotoreflection[1] or rotary reflection[2] is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane including that axis.[3] In 3D, equivalently it is the combination of a rotation and an inversion in a point on the axis.[1] Therefore it is also called a rotoinversion or rotary inversion. A three-dimensional symmetry that has only one fixed point is necessarily an improper rotation.[2]

126 22.1. SEE ALSO 127

In both cases the operations commute. Rotoreflection and rotoinversion are the same if they differ in angle of rotation by 180°, and the point of inversion is in the plane of reflection. An improper rotation of an object thus produces a rotation of its mirror image. The axis is called the rotation- reflection axis.[4] This is called an n-fold improper rotation if the angle of rotation is 360°/n.[4] The notation Sn (S for "Spiegel", German for mirror) denotes the symmetry group generated by an n-fold improper rotation (not to be confused with the same notation for symmetric groups).[4] The notation n¯ is used for n-fold rotoinversion, i.e. rotation by an angle of rotation of 360°/n with inversion. The Coxeter notation for S₂ is [2n+,2+], and orbifold notation is n×. In a wider sense, an “improper rotation” may be defined as any indirect isometry, i.e., an element of E(3)\E+(3) (see Euclidean group): thus it can also be a pure reflection in a plane, or have a glide plane. An indirect isometry is an affine transformation with an orthogonal matrix that has a determinant of −1. A proper rotation is an ordinary rotation. In the wider sense, a “proper rotation” is defined as a direct isometry, i.e., an element of E+(3): it can also be the identity, a rotation with a translation along the axis, or a pure translation. A direct isometry is an affine transformation with an orthogonal matrix that has a determinant of 1. In either the narrower or the wider senses, the composition of two improper rotations is a proper rotation, and the composition of an improper and a proper rotation is an improper rotation. When studying the symmetry of a physical system under an improper rotation (e.g., if a system has a mirror symmetry plane), it is important to distinguish between vectors and pseudovectors (as well as scalars and pseudoscalars, and in general between tensors and pseudotensors), since the latter transform differently under proper and improper rotations (in 3 dimensions, pseudovectors are invariant under inversion).

22.1 See also

• Isometry

• Orthogonal group

22.2 References

[1] Morawiec, Adam (2004), Orientations and Rotations: Computations in Crystallographic Textures, Springer, p. 7, ISBN 9783540407348.

[2] Kinsey, L. Christine; Moore, Teresa E. (2002), Symmetry, Shape, and Surfaces: An Introduction to Mathematics Through Geometry, Springer, p. 267, ISBN 9781930190092.

[3] Salomon, David (1999), Computer Graphics and Geometric Modeling, Springer, p. 84, ISBN 9780387986821.

[4] Bishop, David M. (1993), Group Theory and Chemistry, Courier Dover Publications, p. 13, ISBN 9780486673554. Chapter 23

Inverse function

In mathematics, an inverse function is a function that “reverses” another function. That is, if f is a function mapping x to y, then the inverse function of f maps y back to x.[1]

23.1 Definitions

See also: Inverse element

Let f be a function whose domain is the set X, and whose image (range) is the set Y. Then f is invertible if there exists a function g with domain Y and image X, with the property:

f(x) = y ⇔ g(y) = x.

If f is invertible, the function g is unique, which means that there is exactly one function g satisfying this property (no more, no less). That function g is then called the inverse of f, and is usually denoted as f −1. Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range Y, in which case the inverse relation is the inverse function.[2] Not all functions have an inverse. For this rule to be applicable, each element y ∈ Y must correspond to no more than one x ∈ X; a function f with this property is called one-to-one or an injection. If f and f −1 are functions on X and Y respectively, then both are bijections. The inverse of an injection that is not a bijection is a partial function, that means for some y ∈ Y it is undefined.

23.1.1 Example: squaring and square root functions

The function f(x) = x2 may or may not be invertible, depending on what kinds of numbers are being considered (the “domain”). If the domain is the real numbers, then each possible result y (except 0) corresponds to two different starting points in X – one positive and one negative, and so this function is not invertible: as it is impossible to deduce an input from its output. Such a function is called non-injective or information-losing. If the domain of the function is restricted to the nonnegative reals then the function is injective and invertible.

23.1.2 Inverses in higher mathematics

The definition given above is commonly adopted in set theory and calculus. In higher mathematics, the notation

f : X → Y

128 23.2. PROPERTIES 129

means “f is a function mapping elements of a set X to elements of a set Y ". The source, X, is called the domain of f, and the target, Y, is called the codomain. The codomain contains the range of f as a subset, and is part of the definition of f.[3] When using codomains, the inverse of a function f: X → Y is required to have domain Y and codomain X. For the inverse to be defined on all of Y, every element of Y must lie in the range of the function f. A function with this property is called onto or surjective. Thus, a function with a codomain is invertible if and only if it is both injective (one-to-one) and surjective (onto). Such a function is called a one-to-one correspondence or a bijection, and has the property that every element y ∈ Y corresponds to exactly one element x ∈ X.

23.1.3 Inverses and composition

If f is an invertible function with domain X and range Y, then

f −1 ( f(x) ) = x , for every x ∈ X.

Using the composition of functions we can rewrite this statement as follows:

−1 f ◦ f = idX ,

where idX is the identity function on the set X; that is, the function that leaves its argument unchanged. In category theory, this statement is used as the definition of an inverse morphism. Considering function composition helps to understand the notation f −1. Repeatedly composing a function with itself is called iteration.If f is applied n times, starting with the value x, then this is written as f n(x); so f 2(x) = f (f (x)), etc. Since f −1(f (x)) = x, composing f −1 and f n yields f n−1, “undoing” the effect of one application of f.

23.1.4 Note on notation

Whereas the notation f −1(x) might be misunderstood, f(x)−1 certainly denotes the multiplicative inverse of f(x) and has nothing to do with inversion of f. The expression sin−1 x does not represent the multiplicative inverse to sin x,[4] but the inverse of the sine function applied to x (actually a partial inverse; see below). To avoid confusion, an inverse trigonometric function is often indicated by the prefix "arc" (for Latin arcus). For instance, the inverse of the sine function is typically called the arcsine function, written as arcsin. Similarly, the inverse of a hyperbolic function is indicated by the prefix "ar" (for Latin area).

23.2 Properties

23.2.1 Uniqueness

If an inverse function exists for a given function f, it is unique: it must be the inverse relation.

23.2.2 Symmetry

There is a symmetry between a function and its inverse. Specifically, if f is an invertible function with domain X and range Y, then its inverse f −1 has domain Y and range X, and the inverse of f −1 is the original function f. In symbols, for functions f:X→Y and g:Y→X,

g ◦ f = idX ⇒ f ◦ g = idY .

This follows from the connection between function inverse and relation inverse, because inversion of relations is an involution. 130 CHAPTER 23. INVERSE FUNCTION

This statement is an obvious consequence of the deduction that for f to be invertible it must be injective (first definition of the inverse) or bijective (second definition). The property of involutive symmetry can be concisely expressed by the following formula:

( )−1 f −1 = f.

The inverse of a composition of functions is given by the formula

(g ◦ f)−1 = f −1 ◦ g−1

Notice that the order of g and f have been reversed; to undo f followed by g, we must first undo g and then undo f. For example, let f(x) = 3x and let g(x) = x + 5. Then the composition g ∘ f is the function that first multiplies by three and then adds five:

(g ◦ f)(x) = 3x + 5

To reverse this process, we must first subtract five, and then divide by three:

◦ −1 1 − (g f) (y) = 3 (y 5) This is the composition (f −1 ∘ g −1)(y).

23.2.3 Self-inverses

If X is a set, then the identity function on X is its own inverse:

−1 idX = idX

More generally, a function f : X → X is equal to its own inverse if and only if the composition f ∘ f is equal to idX. Such a function is called an involution.

23.3 Inverses in calculus

Single-variable calculus is primarily concerned with functions that map real numbers to real numbers. Such functions are often defined through formulas, such as: f(x) = (2x + 8)3.

A function f from the real numbers to the real numbers possesses an inverse as long as it is one-to-one, i.e. as long as the graph of y = f(x) has, for each possible y value only one corresponding x value, and thus passes the horizontal line test. The following table shows several standard functions and their inverses:

23.3.1 Formula for the inverse

One approach to finding a formula for f −1, if it exists, is to solve the equation y = f(x) for x. For example, if f is the function 23.3. INVERSES IN CALCULUS 131

f(x) = (2x + 8)3

then we must solve the equation y = (2x + 8)3 for x:

y = (2x + 8)3 √ 3 y = 2x + 8 √ 3 y − 8 = 2x √ 3 y − 8 = x. 2 Thus the inverse function f −1 is given by the formula

√ 3 y − 8 f −1(y) = . 2 Sometimes the inverse of a function cannot be expressed by a formula with a finite number of terms. For example, if f is the function

f(x) = x − sin x,

then f is one-to-one, and therefore possesses an inverse function f −1. The formula for this inverse has an infinite number of terms:

( ( )) ∑∞ n n−1 n −1 y 3 d √ θ f (y) = lim − n! θ→0 dθ n 1 3 − n=1 θ sin(θ)

23.3.2 Graph of the inverse

If f is invertible, then the graph of the function

y = f −1(x)

is the same as the graph of the equation

x = f(y).

This is identical to the equation y = f(x) that defines the graph of f, except that the roles of x and y have been reversed. Thus the graph of f −1 can be obtained from the graph of f by switching the positions of the x and y axes. This is equivalent to reflecting the graph across the line y = x.

23.3.3 Inverses and derivatives

A continuous function f is one-to-one (and hence invertible) if and only if it is either strictly increasing or decreasing (with no local maxima or minima). For example, the function

f(x) = x3 + x 132 CHAPTER 23. INVERSE FUNCTION

is invertible, since the derivative f′(x) = 3x2 + 1 is always positive. If the function f is differentiable, then the inverse f −1 will be differentiable as long as f′(x) ≠ 0. The derivative of the inverse is given by the inverse function theorem:

( )′ 1 f −1 (y) = . f ′ (f −1(y))

If we set y = f(x), then the formula above can be written

dx 1 = . dy dy/dx

This result follows from the chain rule (see the article on inverse functions and differentiation). The inverse function theorem can be generalized to functions of several variables. Specifically, a differentiable multivariable function f : Rn → Rn is invertible in a neighborhood of a point p as long as the Jacobian matrix of f at p is invertible. In this case, the Jacobian of f −1 at f(p) is the matrix inverse of the Jacobian of f at p.

23.4 Real-world examples

1. Let f be the function that converts a temperature in degrees Celsius to a temperature in degrees Fahrenheit:

9 F = f(C) = 5 C + 32;

then its inverse function converts degrees Fahrenheit to degrees Celsius:

−1 5 − C = f (F ) = 9 (F 32),

since

( ) (( ) ) −1 −1 9 5 9 − f ( f(C) ) = f 5 C + 32 = 9 5 C + 32 32 = C every for ,C.

2. Suppose f assigns each child in a family its birth year. An inverse function would output which child was born in a given year. However, if the family has twins (or triplets) then the output cannot be known when the input is the common birth year. As well, if a year is given in which no child was born then a child cannot be named. But if each child was born in a separate year, and if we restrict attention to the three years in which a child was born, then we do have an inverse function. For example,

f(Allan) = 2005, f(Brad) = 2007, f(Cary) = 2001 f −1(2005) = Allan, f −1(2007) = Brad, f −1(2001) = Cary

3. Let R be the function that leads to an x percentage rise of some quantity, and F be the function producing an x percentage fall. Applied to $100 with x = 10%, we find that applying the first function followed by the second does not restore the original value of $100, demonstrating the fact that, despite appearances, these two functions are not inverses of each other.

23.5 Generalizations 23.5. GENERALIZATIONS 133

23.5.1 Partial inverses

Even if a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain. For example, the function

f(x) = x2

is not one-to-one, since x2 = (−x)2. However, the function becomes one-to-one if we restrict to the domain x ≥ 0, in which case

√ f −1(y) = y.

(If we instead restrict to the domain x ≤ 0, then the inverse is the negative of the square root of y.) Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function:

√ f −1(y) =  y.

Sometimes this multivalued inverse is called the full inverse of f, and the portions (such as √x and −√x) are called branches. The most important branch of a multivalued function (e.g. the positive square root) is called the principal branch, and its value at y is called the principal value of f −1(y). For a continuous function on the real line, one branch is required between each pair of local extrema. For example, the inverse of a cubic function with a local maximum and a local minimum has three branches (see the picture to the right). These considerations are particularly important for defining the inverses of trigonometric functions. For example, the sine function is not one-to-one, since

sin(x + 2π) = sin(x)

for every real x (and more generally sin(x + 2πn) = sin(x) for every integer n). However, the sine is one-to-one on the interval [−π/2, π/2], and the corresponding partial inverse is called the arcsine. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between −π/2 and π/2. The following table describes the principal branch of each inverse trigonometric function:

23.5.2 Left and right inverses

If f: X → Y, a left inverse for f (or retraction of f) is a function g: Y → X such that

g ◦ f = idX .

That is, the function g satisfies the rule

If f(x) = y , then g(y) = x.

Thus, g must equal the inverse of f on the image of f, but may take any values for elements of Y not in the image. A function f with a left inverse is necessarily injective. In classical mathematics, every injective function f necessarily has a left inverse; however, this may fail in constructive mathematics. For instance, a left inverse of the inclusion {0,1} → R of the two-element set in the reals violates indecomposability by giving a retraction of the real line to the set {0,1} . A right inverse for f (or section of f) is a function h: Y → X such that 134 CHAPTER 23. INVERSE FUNCTION

f ◦ h = idY . That is, the function h satisfies the rule

If h(y) = x , then f(x) = y.

Thus, h(y) may be any of the elements of X that map to y under f. A function f has a right inverse if and only if it is surjective (though constructing such an inverse in general requires the axiom of choice). An inverse which is both a left and right inverse must be unique. Likewise, if g is a left inverse for f, then g may or may not be a right inverse for f; and if g is a right inverse for f, then g is not necessarily a left inverse for f. For example let f: R → [0, ∞) denote the squaring map, such that f(x) = x2 for all x in R, and let g: [0, ∞) → R denote the square root map, such that g(x) = √x for all x ≥ 0. Then f(g(x)) = x for all x in [0, ∞); that is, g is a right inverse to f. However, g is not a left inverse to f, since, e.g., g(f(−1)) = 1 ≠ −1.

23.5.3 Preimages

If f: X → Y is any function (not necessarily invertible), the preimage (or inverse image) of an element y ∈ Y is the set of all elements of X that map to y: f −1({y}) = {x ∈ X : f(x) = y} . The preimage of y can be thought of as the image of y under the (multivalued) full inverse of the function f. Similarly, if S is any subset of Y, the preimage of S is the set of all elements of X that map to S:

f −1(S) = {x ∈ X : f(x) ∈ S} . For example, take a function f: R → R, where f: x ↦ x2. This function is not invertible for reasons discussed above. Yet preimages may be defined for subsets of the codomain:

f −1({1, 4, 9, 16}) = {−4, −3, −2, −1, 1, 2, 3, 4} The preimage of a single element y ∈ Y – a singleton set {y} – is sometimes called the fiber of y. When Y is the set of real numbers, it is common to refer to f −1({y}) as a level set.

23.6 See also

• Inverse function theorem, gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain and gives a formula for the derivative of the inverse function • Inverse functions and differentiation • Inverse relation • Lagrange inversion theorem, gives the Taylor series expansion of the inverse function of an analytic function

23.7 Notes

[1] Keisler, H. Jerome. “Differentiation” (PDF). Retrieved 2015-01-24. § 2.4 [2] Smith, Eggen & St. Andre 2006, p. 202, Theorem 4.9 [3] Smith, Eggen & St. Andre 2006, p. 179 [4] Thomas 1972, pp. 304-309 23.8. REFERENCES 135

23.8 References

• Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006), A Transition to Advanced Mathematics (6th ed.), Thompson Brooks/Cole, ISBN 978-0-534-39900-9 • Thomas, Jr., George B. (1972), Calculus and Analytic Geometry Part 1: Functions of One Variable and Analytic Geometry (Alternate ed.), Addison-Wesley

23.9 Further reading

• Spivak, Michael (1994), Calculus (3rd ed.), Publish or Perish, ISBN 0-914098-89-6 • Stewart, James (2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0-534-39339-7

23.10 External links

• Hazewinkel, Michiel, ed. (2001), “Inverse function”, Encyclopedia of Mathematics, Springer, ISBN 978-1- 55608-010-4

• Wikibook: Functions • Wolfram Mathworld: Inverse Function 136 CHAPTER 23. INVERSE FUNCTION

A function f and its inverse f −1. Because f maps a to 3, the inverse f −1 maps 3 back to a. 23.10. EXTERNAL LINKS 137

If f maps X to Y, then f −1 maps Y back to X.

The inverse of g ∘ f is f −1 ∘ g −1. 138 CHAPTER 23. INVERSE FUNCTION

The graphs of y = f(x) and y = f −1(x). The dotted line is y = x. 23.10. EXTERNAL LINKS 139

y y = x2

y = √x̅

x

x = y

The square root of x is a partial inverse to f(x) = x2. 140 CHAPTER 23. INVERSE FUNCTION

The inverse of this cubic function has three branches. 23.10. EXTERNAL LINKS 141

The arcsine is a partial inverse of the sine function. Chapter 24

Linear function (calculus)

Not to be confused with linear map. In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates with uniform scales) is a line in the plane.[1] Their characteristic property that when the value of the input variable is changed, the change in the output is a constant multiple of the change in the input variable. Linear functions are related to linear equations.

24.1 Properties

A linear function is a polynomial function in which the variable x has degree at most one, which means it is of the form[2]

f(x) = ax + b.

Here x is the variable. The graph of a linear function, that is, the set of all points whose coordinates have the form (x, f(x)), is a line on the Cartesian plane (if over real numbers). That is why this type of function is called linear. Some authors, for various reasons, also require that the coefficient of the variable (the a in ax + b) should not be zero.[3] This requirement can also be expressed by saying that the degree of the polynomial defining the function is exactly one, or by saying that the line which is the graph of a linear function is a slanted line (neither vertical nor horizontal). This requirement will not be imposed in this article, thus constant functions, f(x) = b, will be considered to be linear functions (their graphs are horizontal lines). The domain or set of allowed values for x of a linear function is the entire set of real numbers R, or whatever field that is in use. This means that any (real) number can be substituted for x. Because two different points determine a line, it is enough to substitute two different values for x in the linear function and determine f(x) for each of these values. This will give the coordinates of two different points that lie on the line. Because f is a function, this line will not be vertical. If the value of either or both of the coefficient letters a and b are changed, a different line is obtained. Since the graph of a linear function is a nonvertical line, this line has exactly one intersection point with the y-axis. This point is (0, b). The graph of a nonconstant linear function has exactly one intersection point with the x-axis. This point is (−b/a, 0). From this, it follows that a nonconstant linear function has exactly one zero or root. That is, there is exactly one solution to the equation ax + b = 0. The zero is x = −b/a.

24.2 Slope

The slope of a nonvertical line is a number that measures how steeply the line is slanted. The first derivative of a linear function, in the sense of calculus, is exactly this slope of the graph of the function. For f(x) = ax + b, this slope

142 24.3. RELATIONSHIP WITH LINEAR EQUATIONS 143

Graph of the linear function: y(x) = −x + 2

and derivative is given by the constant a. Linear functions can be characterized as the only real-valued functions that are defined on the entire real line and have a constant derivative. The derivative of a function, in general, measures its rate of change. Because a linear function f(x) = ax + b has a constant rate of change a, it has the property that whenever the input x is increased by one unit, the output changes by a units. If a is positive, this will cause the value of the function to increase, while if a is negative it will cause the value to decrease. More generally, if the input increases by some other amount, c, the output changes by ca.

24.3 Relationship with linear equations

The points on a line have coordinates which can also be thought of as the solutions of linear equations in two variables (the equation of the line). These solution sets define functions which are linear functions. This connection between linear equations and linear functions provides the most common way to produce linear functions. The equation y = ax + b is referred to as the slope-intercept form of a linear equation. In this form, the variable is x, and y, is the value of the function. It also has two coefficients, a and b. In this instance, the fact that the values of y depend on the values of x is an expression of the functional relationship between them. To be very explicit, the 144 CHAPTER 24. LINEAR FUNCTION (CALCULUS)

Δy

Δx

The slope of a line is the ratio between a change in x, denoted Δx, and the corresponding change in y, denoted Δy

linear equation is expressing the equality of values of the dependent variable y with the functional values of the linear function f(x) = ax + b, in other words y = f(x) for this particular linear function f. If the linear function f is given, the linear equation of the graph of this function is obtained by defining the variable y to be the functional value f(x), that is, setting y = f(x) = ax + b and suppressing the functional notation in the middle. Starting with a linear equation, one can create linear functions, but this is a more subtle operation and must be done with care. Why this is so is not immediately apparent when the linear equation has the slope-intercept form, so this discussion will be postponed. For the moment observe that if the linear equation has the slope-intercept form, then 24.3. RELATIONSHIP WITH LINEAR EQUATIONS 145

the expression that the dependent variable y is equal to is the linear function whose graph is the line satisfying the linear equation. The constant b is the so-called y-intercept. It is the y-value at which the line intersects the y-axis. The coefficient a is the slope of the line. This measures of the rate of change of the linear function associated with the line. Since a is a constant, this rate of change is constant. Moving from any point on the line to the right by one unit (that is, increasing x by 1), the y-value of the point’s coordinate changes by a. This is expressed functionally by the statement that f(x + 146 CHAPTER 24. LINEAR FUNCTION (CALCULUS)

1) = f(x) + a when f(x) = ax + b. For example, the slope-intercept form y = −2x + 4 has a = −2 and b = 4. The point (0, b) = (0, 4) is the intersection of the line and the y-axis, the point (−b/a, 0) = (−4/−2, 0) = (2, 0) is the intersection of the line and the x-axis, and a = −2 is the slope of the line. For every step to the right (x increases by 1), the value of y changes by −2 (goes down).

If the linear equation in the general form

Ax + By = C.

has B ≠ 0, then it may be solved for the variable y and thus used to define a linear function, namely, y = −(A/B)x + (C/B) = f(x). While all lines have equations in the general form, only the non-vertical lines have equations which can give rise to linear functions.

24.4 Relationship with other classes of functions

If the coefficient of the variable is not zero (a ≠ 0), then a linear function is represented by a degree 1 polynomial (also called a linear polynomial), otherwise it is a constant function – also a polynomial function, but of zero degree. A straight line, when drawn in a different kind of coordinate system may represent other functions. For example, it may represent an exponential function when its values are expressed in the logarithmic scale. It means that when log(g(x)) is a linear function of x, the function g is exponential. With linear functions, increasing the input 24.5. NOTES 147 by one unit causes the output to increase by a fixed amount, which is the slope of the graph of the function. With exponential functions, increasing the input by one unit causes the output to increase by a fixed multiple, which is known as the base of the exponential function. If both arguments and values of a function are in the logarithmic scale (i.e., when log(y) is a linear function of log(x)), then the straight line represents a power law:

⇒ b · a logr y = a logr x + b y = r x On the other hand, the graph of a linear function in terms of polar coordinates: r = f(φ) = aφ + b is an Archimedean spiral if a ≠ 0 and a circle otherwise.

24.5 Notes

[1] Stewart 2012, p. 23

[2] Stewart 2012, p. 24

[3] Swokowski 1983, p. 34 is but one of many well known references that could be cited.

24.6 See also

• Affine map, a generalization

• Arithmetic progression, a linear function of integer argument

24.7 References

• James Stewart (2012), Calculus: Early Transcendentals, edition 7E, Brooks/Cole. ISBN 978-0-538-49790-9 • Swokowski, Earl W. (1983), Calculus with analytic geometry (Alternate ed.), Boston: Prindle, Weber & Schmidt, ISBN 0871503417

24.8 External links

• http://www.math.okstate.edu/~{}noell/ebsm/linear.html

• http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf Chapter 25

Linear map

“Linear transformation” redirects here. For fractional linear transformations, see Möbius transformation. Not to be confused with linear function.

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. Linear maps can generally be represented as matrices, and simple examples include rotation and reflection linear transformations. An important special case is when V = W, in which case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the same meaning as linear map, while in analytic geometry it does not. A linear map always maps linear subspaces onto linear subspaces (possibly of a lower dimension); for instance it maps a plane through the origin to a plane, straight line or point. In the language of abstract algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring.

25.1 Definition and first consequences

Let V and W be vector spaces over the same field K. A function f : V → W is said to be a linear map if for any two vectors x and y in V and any scalar α in K, the following two conditions are satisfied:

This is equivalent to requiring the same for any linear combination of vectors, i.e. that for any vectors x1, ..., xm ∈ V and scalars a1, ..., am ∈ K, the following equality holds:

f(a1x1 + ··· + amxm) = a1f(x1) + ··· + amf(xm).

Denoting the zero elements of the vector spaces V and W by 0V and 0W respectively, it follows that f(0V) = 0W because letting α = 0 in the equation for homogeneity of degree 1,

f(0V ) = f(0 · 0V ) = 0 · f(0V ) = 0W .

Occasionally, V and W can be considered to be vector spaces over different fields. It is then necessary to specify which of these ground fields is being used in the definition of “linear”. If V and W are considered as spaces over the field K as above, we talk about K-linear maps. For example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. A linear map from V to K (with K viewed as a vector space over itself) is called a linear functional. These statements generalize to any left-module RM over a ring R without modification, and to any right-module upon reversing of the scalar multiplication.

148 25.2. EXAMPLES 149

25.2 Examples

• The zero map between two left-modules (or two right-modules) over the same ring is always linear.

• The identity map on any module is a linear operator.

• Any homothecy centered in the origin of a vector space, v 7→ cv where c is a scalar, is a linear operator. Generalization of this statement to modules is more complicated.

• For real numbers, the map x ↦ x2 is not linear.

• For real numbers, the map x ↦ x + 1 is not linear (but is an affine transformation; y = x + 1 is a linear equation, as the term is used in analytic geometry.)

• If A is a real m × n matrix, then A defines a linear map from Rn to Rm by sending the column vector x ∈ Rn to the column vector Ax ∈ Rm. Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner; see the following section.

• Differentiation defines a linear map from the space of all differentiable functions to the space of all functions. It also defines a linear operator on the space of all smooth functions.

• The (definite) integral over some interval I is a linear map from the space of all real-valued integrable functions on I to R

• The (indefinite) integral (or antiderivative) with a fixed starting point defines a linear map from the space of all real-valued integrable functions on R to the space of all real-valued, differentiable, functions on R. Without a fixed starting point, an exercise in group theory will show that the antiderivative maps to the quotient group of the differentiables over( the∫ equivalence relation, “differ) by a constant”, which yields an identity class of the constant valued functions : I(ℜ) → D(ℜ)/ℜ .

• If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f : V → W to dimF(W) × dimF(V) matrices in the way described in the sequel are themselves linear maps (indeed linear isomorphisms).

• The expected value of a random variable (which is in fact a function, and as such a member of a vector space) is linear, as for random variables X and Y we have E[X + Y] = E[X] + E[Y] and E[aX] = aE[X], but the variance of a random variable is not linear.

25.3 Matrices

Main article: Transformation matrix

If V and W are finite-dimensional vector spaces and a basis is defined for each vector space, then every linear map from V to W can be represented by a matrix. This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if A is a real m × n matrix, then f(x) = Ax describes a linear map Rn → Rm (see Euclidean space).

Let {v1, ..., vn} be a basis for V. Then every vector v in V is uniquely determined by the coefficients c1, ..., cn in the field R:

c1v1 + ··· + cnvn.

If f : V → W is a linear map,

f(c1v1 + ··· + cnvn) = c1f(v1) + ··· + cnf(vn), which implies that the function f is entirely determined by the vectors f(v1), ..., f(vn). Now let {w1, ..., wm} be a basis for W. Then we can represent each vector f(vj) as 150 CHAPTER 25. LINEAR MAP

f(vj) = a1jw1 + ··· + amjwm.

Thus, the function f is entirely determined by the values of aij. If we put these values into an m × n matrix M, then we can conveniently use it to compute the vector output of f for any vector in V. To get M, every column j of M is a vector

T (a1j, ..., amj)

corresponding to f(vj) as defined above. To define it more clearly, for some column j that corresponds to the mapping f(vj),

  a1j    .    M =  ∗ . ∗  .  amj where M is the matrix of f. The symbol ∗ denotes that there are other columns which together with column j make up a total of n columns of M. In other words, every column j = 1, ..., n has a corresponding vector f(vj) whose coordinates a₁j, ..., amj are the elements of column j. A single linear map may be represented by many matrices. This is because the values of the elements of a matrix depend on the bases chosen. The matrices of a linear transformation can be represented visually:

1. Matrix for T relative to B : A

2. Matrix for T relative to B′ : A′

3. Transition matrix from B′ to B : P

4. Transition matrix from B to B′ : P −1 25.4. EXAMPLES OF LINEAR TRANSFORMATION MATRICES 151

Such that starting in the bottom left corner [⃗v]B′ and looking for the bottom right corner [T (⃗v)]B′ , one would left- ′ multiply—that is, A [⃗v]B′ = [T (⃗v)]B′ . The equivalent method would be the “longer” method going clockwise from −1 −1 the same point such that [⃗v]B′ is left-multiplied with P AP , or P AP [⃗v]B′ = [T (⃗v)]B′ .

25.4 Examples of linear transformation matrices

In two-dimensional space R2 linear maps are described by 2 × 2 real matrices. These are some examples:

• rotation by 90 degrees counterclockwise: ( ) 0 −1 A = 1 0

• rotation by angle θ counterclockwise: ( ) cos θ − sin θ A = sin θ cos θ

• reflection against the x axis: ( ) 1 0 A = 0 −1

• reflection against the y axis: ( ) −1 0 A = 0 1

• scaling by 2 in all directions: ( ) 2 0 A = 0 2 152 CHAPTER 25. LINEAR MAP

• horizontal shear mapping: ( ) 1 m A = 0 1

• squeeze mapping: ( ) k 0 A = 0 1/k

• projection onto the y axis: ( ) 0 0 A = . 0 1

25.5 Forming new linear maps from given ones

The composition of linear maps is linear: if f : V → W and g : W → Z are linear, then so is their composition g ∘ f : V → Z. It follows from this that the class of all vector spaces over a given field K, together with K-linear maps as morphisms, forms a category. The inverse of a linear map, when defined, is again a linear map.

If f1 : V → W and f2 : V → W are linear, then so is their pointwise sum f1 + f2 (which is defined by (f1 + f2)(x) = f1(x) + f2(x)). If f : V → W is linear and a is an element of the ground field K, then the map af, defined by (af)(x) = a(f(x)), is also linear. Thus the set L(V, W) of linear maps from V to W itself forms a vector space over K, sometimes denoted Hom(V, W). Furthermore, in the case that V = W, this vector space (denoted End(V)) is an associative algebra under composition of maps, since the composition of two linear maps is again a linear map, and the composition of maps is always associative. This case is discussed in more detail below. Given again the finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the matrix multiplication, the addition of linear maps corresponds to the matrix addition, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.

25.6 Endomorphisms and automorphisms

Main articles: Endomorphism and Automorphism

A linear transformation f: V → V is an endomorphism of V; the set of all such endomorphisms End(V) together with addition, composition and scalar multiplication as defined above forms an associative algebra with identity element over the field K (and in particular a ring). The multiplicative identity element of this algebra is the identity map id: V → V. An endomorphism of V that is also an isomorphism is called an automorphism of V. The composition of two auto- morphisms is again an automorphism, and the set of all automorphisms of V forms a group, the automorphism group of V which is denoted by Aut(V) or GL(V). Since the automorphisms are precisely those endomorphisms which possess inverses under composition, Aut(V) is the group of units in the ring End(V). If V has finite dimension n, then End(V) is isomorphic to the associative algebra of all n × n matrices with entries in K. The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n × n invertible matrices with entries in K.

25.7 Kernel, image and the rank–nullity theorem

Main articles: Kernel (linear operator), Image (mathematics) and Rank of a matrix 25.8. COKERNEL 153

If f : V → W is linear, we define the kernel and the image or range of f by

ker(f) = { x ∈ V : f(x) = 0 } im(f) = { w ∈ W : w = f(x), x ∈ V } ker(f) is a subspace of V and im(f) is a subspace of W. The following dimension formula is known as the rank–nullity theorem:

dim(ker(f)) + dim(im(f)) = dim(V ). The number dim(im(f)) is also called the rank of f and written as rank(f), or sometimes, ρ(f); the number dim(ker(f)) is called the nullity of f and written as null(f) or ν(f). If V and W are finite-dimensional, bases have been chosen and f is represented by the matrix A, then the rank and nullity of f are equal to the rank and nullity of the matrix A, respectively.

25.8 Cokernel

Main article: Cokernel

A subtler invariant of a linear transformation is the cokernel, which is defined as

coker f := W /f(V ) = W /im(f). This is the dual notion to the kernel: just as the kernel is a subspace of the domain, the co-kernel is a quotient space of the target. Formally, one has the exact sequence

0 → ker f → V → W → coker f → 0. These can be interpreted thus: given a linear equation f(v) = w to solve,

• the kernel is the space of solutions to the homogeneous equation f(v) = 0, and its dimension is the number of degrees of freedom in a solution, if it exists; • the co-kernel is the space of constraints that must be satisfied if the equation is to have a solution, and its dimension is the number of constraints that must be satisfied for the equation to have a solution.

The dimension of the co-kernel and the dimension of the image (the rank) add up to the dimension of the target space. For finite dimensions, this means that the dimension of the quotient space W/f(V) is the dimension of the target space minus the dimension of the image. As a simple example, consider the map f: R2 → R2, given by f(x, y) = (0, y). Then for an equation f(x, y) = (a, b) to have a solution, we must have a = 0 (one constraint), and in that case the solution space is (x, b) or equivalently stated, (0, b) + (x, 0), (one degree of freedom). The kernel may be expressed as the subspace (x, 0) < V: the value of x is the freedom in a solution – while the cokernel may be expressed via the map W → R, (a, b) 7→ (a): given a vector (a, b), the value of a is the obstruction to there being a solution. ∞ ∞ An example illustrating the infinite-dimensional case is afforded by the map f: R → R , {an} 7→ {bn} with b1 = 0 and bn ₊ ₁ = an for n > 0. Its image consists of all sequences with first element 0, and thus its cokernel consists of the classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its co-kernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same sum as the rank and the dimension of the co-kernel ( ℵ0 + 0 = ℵ0 + 1 ), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an endomorphism have the same dimension (0 ≠ 1). The reverse situation obtains for the map h: R∞ → R∞, {an} 7→ {cn} with cn = an ₊ ₁. Its image is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1. 154 CHAPTER 25. LINEAR MAP

25.8.1 Index

For a linear operator with finite-dimensional kernel and co-kernel, one may define index as:

ind f := dim ker f − dim coker f, namely the degrees of freedom minus the number of constraints. For a transformation between finite-dimensional vector spaces, this is just the difference dim(V) − dim(W), by rank– nullity. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints. Con- versely, if mapping from a smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom. The index of an operator is precisely the Euler characteristic of the 2-term complex 0 → V → W → 0. In operator theory, the index of Fredholm operators is an object of study, with a major result being the Atiyah–Singer index theorem.[1]

25.9 Algebraic classifications of linear transformations

No classification of linear maps could hope to be exhaustive. The following incomplete list enumerates some impor- tant classifications that do not require any additional structure on the vector space. Let V and W denote vector spaces over a field, F. Let T: V → W be a linear map.

• T is said to be injective or a monomorphism if any of the following equivalent conditions are true:

• T is one-to-one as a map of sets. • kerT = {0V} • T is monic or left-cancellable, which is to say, for any vector space U and any pair of linear maps R: U → V and S: U → V, the equation TR = TS implies R = S. • T is left-invertible, which is to say there exists a linear map S: W → V such that ST is the identity map on V.

• T is said to be surjective or an epimorphism if any of the following equivalent conditions are true:

• T is onto as a map of sets. • coker T = {0W} • T is epic or right-cancellable, which is to say, for any vector space U and any pair of linear maps R: W → U and S: W → U, the equation RT = ST implies R = S. • T is right-invertible, which is to say there exists a linear map S: W → V such that TS is the identity map on W.

• T is said to be an isomorphism if it is both left- and right-invertible. This is equivalent to T being both one-to- one and onto (a bijection of sets) or also to T being both epic and monic, and so being a bimorphism.

• If T: V → V is an endomorphism, then:

• If, for some positive integer n, the n-th iterate of T, Tn, is identically zero, then T is said to be nilpotent. • If T2 = T, then T is said to be idempotent • If T = kI, where k is some scalar, then T is said to be a scaling transformation or scalar multiplication map; see scalar matrix. 25.10. CHANGE OF BASIS 155

25.10 Change of basis

Main articles: Basis (linear algebra) and Change of basis

Given a linear map whose matrix is A, in the basis B of the space it transforms vectors coordinates [u] as [v] = A[u]. As vectors change with the inverse of B, its inverse transformation is [v] = B[v']. Substituting this in the first expression

B[v′] = AB[u′] hence

[v′] = B−1AB[u′] = A′[u′]. Therefore the matrix in the new basis is A′ = B−1AB, being B the matrix of the given basis. Therefore linear maps are said to be 1-co 1-contra -variant objects, or type (1, 1) tensors.

25.11 Continuity

Main article: Discontinuous linear map

A linear transformation between topological vector spaces, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded, for example, when the domain is finite-dimensional. An infinite-dimensional domain may have discontinuous linear operators. An example of an unbounded, hence discontinuous, linear transformation is differentiation on the space of smooth functions equipped with the supremum norm (a function with small values can have a derivative with large values, while the derivative of 0 is 0). For a specific example, sin(nx)/n converges to 0, but its derivative cos(nx) does not, so differentiation is not continuous at 0 (and by a variation of this argument, it is not continuous anywhere).

25.12 Applications

A specific application of linear maps is for geometric transformations, such as those performed in computer graph- ics, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix. Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to derivatives; or in relativity, used as a device to keep track of the local transformations of reference frames. Another application of these transformations is in compiler optimizations of nested-loop code, and in parallelizing compiler techniques.

25.13 See also

• Affine transformation • Linear equation • Bounded operator • Antilinear map • Semilinear transformation • Continuous linear operator • Bent function 156 CHAPTER 25. LINEAR MAP

25.14 Notes

[1] Nistor, Victor (2001), “Index theory”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608- 010-4: “The main question in index theory is to provide index formulas for classes of Fredholm operators ... Index theory has become a subject on its own only after M.F. Atiyah and I. Singer published their index theorems”

25.15 References

• Halmos, Paul R. (1974), Finite-dimensional vector spaces, New York: Springer-Verlag, ISBN 978-0-387- 90093-3 • Lang, Serge (1987), Linear algebra, New York: Springer-Verlag, ISBN 978-0-387-96412-6 Chapter 26

Matrix (mathematics)

For other uses, see Matrix. “Matrix theory” redirects here. For the physics topic, see Matrix string theory. In mathematics, a matrix (plural matrices) is a rectangular array[1]—of numbers, symbols, or expressions, arranged

ai,j n columns j changes m rows . . a1,1 a1,2 a1,3 . i c . . a2,1 . h a2,2 a2,3 a n . . a3,1 a3,2 a3,3 . g e . . . . s ......

Each element of a matrix is often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A.

in rows and columns[2][3]—that is interpreted and manipulated in certain prescribed ways. One such way is to state the order of the matrix. For example, the order of the matrix below is 2 × 3 (read “two by three”), because there are two rows and three columns.

157 158 CHAPTER 26. MATRIX (MATHEMATICS)

[ ] 1 9 −13 20 5 −6

The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix’s eigenvalues and eigenvectors. Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phe- nomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions. A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function.

26.1 Definition

A matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined.[6] Most commonly, a matrix over a field F is a rectangular array of scalars each of which is a member of F.[7][8] Most of this article focuses on real and complex matrices, i.e., matrices whose elements are real numbers or complex numbers, respectively. More general types of entries are discussed below. For instance, this is a real matrix:

  −1.3 0.6 A =  20.4 5.5 . 9.7 −6.2

The numbers, symbols or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.

26.1.1 Size

The size of a matrix is defined by the number of rows and columns that it contains. A matrix with m rows and n columns is called an m × n matrix or m-by-n matrix, while m and n are called its dimensions. For example, the matrix A above is a 3 × 2 matrix. Matrices which have a single row are called row vectors, and those which have a single column are called column vectors. A matrix which has the same number of rows and columns is called a square matrix. A matrix with an infinite number of rows or columns (or both) is called an infinite matrix. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix. 26.2. NOTATION 159

26.2 Notation

Matrices are commonly written in box brackets or large parentheses:

    a11 a12 ··· a1n a11 a12 ··· a1n     a21 a22 ··· a2n a21 a22 ··· a2n     m×n A =  . . . .  =  . . . .  ∈ R .  . . .. .   . . .. .  am1 am2 ··· amn am1 am2 ··· amn The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are usually symbolized using upper-case letters (such as A in the examples above), while the corresponding lower-case letters, with two subscript indices (e.g., a11, or a₁,₁), represent the entries. In addition to using upper-case letters to symbolize matrices, many authors use a special typographical style, commonly boldface upright (non-italic), to further distinguish matrices from other mathematical objects. An alternative notation involves the use of a double-underline with the variable name, with or without boldface style, (e.g., A ). The entry in the i-th row and j-th column of a matrix A is sometimes referred to as the i,j,(i,j), or (i,j)th entry of the matrix, and most commonly denoted as ai,j, or aij. Alternative notations for that entry are A[i,j] or Ai,j. For example, the (1,3) entry of the following matrix A is 5 (also denoted a13, a₁,₃, A[1,3] or A1,3):

  4 −7 5 0 A = −2 0 11 8  19 1 −3 12 Sometimes, the entries of a matrix can be defined by a formula such as ai,j = f(i, j). For example, each of the entries of the following matrix A is determined by aij = i − j.

  0 −1 −2 −3 A = 1 0 −1 −2 2 1 0 −1 In this case, the matrix itself is sometimes defined by that formula, within square brackets or double parenthesis. For example, the matrix above is defined as A = [i-j], or A = ((i-j)). If matrix size is m × n, the above-mentioned formula f(i, j) is valid for any i = 1, ..., m and any j = 1, ..., n. This can be either specified separately, or using m × n as a subscript. For instance, the matrix A above is 3 × 4 and can be defined as A = [i − j](i = 1, 2, 3; j = 1, ..., 4), or A = [i − j]3×4. Some programming languages utilize doubly subscripted arrays (or arrays of arrays) to represent an m-×-n matrix. Some programming languages start the numbering of array indexes at zero, in which case the entries of an m-by-n matrix are indexed by 0 ≤ i ≤ m − 1 and 0 ≤ j ≤ n − 1.[9] This article follows the more common convention in mathematical writing where enumeration starts from 1. The set of all m-by-n matrices is denoted 필(m, n).

26.3 Basic operations

There are a number of basic operations that can be applied to modify matrices, called matrix addition, scalar multi- plication, transposition, matrix multiplication, row operations, and submatrix.[11]

26.3.1 Addition, scalar multiplication and transposition

Main articles: Matrix addition, Scalar multiplication and Transpose

Familiar properties of numbers extend to these operations of matrices: for example, addition is commutative, i.e., the matrix sum does not depend on the order of the summands: A + B = B + A.[12] The transpose is compatible with addition and scalar multiplication, as expressed by (cA)T = c(AT) and (A + B)T = AT + BT. Finally, (AT)T = A. 160 CHAPTER 26. MATRIX (MATHEMATICS)

26.3.2 Matrix multiplication

Main article: Matrix multiplication Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the B

b1,1 b1,2 b1,3

b2,1 b2,2 b2,3

a1,1 a1,2

a a A 2,1 2,2 a3,1 a3,2

a4,1 a4,2

Schematic depiction of the matrix product AB of two matrices A and B.

number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: ∑ ··· n [AB]i,j = Ai,1B1,j + Ai,2B2,j + + Ai,nBn,j = r=1 Ai,rBr,j ,

where 1 ≤ i ≤ m and 1 ≤ j ≤ p.[13] For example, the underlined entry 2340 in the product is calculated as (2 × 1000) + (3 × 100) + (4 × 10) = 2340:

  [ ] 0 1000 [ ] 2 3 4 3 2340 1 100  = . 1 0 0 0 1000 0 10 Matrix multiplication satisfies the rules (AB)C = A(BC)(associativity), and (A+B)C = AC+BC as well as C(A+B) = CA+CB (left and right distributivity), whenever the size of the matrices is such that the various products are defined.[14] The product AB may be defined without BA being defined, namely if A and B are m-by-n and n-by-k matrices, respectively, and m ≠ k. Even if both products are defined, they need not be equal, i.e., generally

AB ≠ BA, 26.4. LINEAR EQUATIONS 161 i.e., matrix multiplication is not commutative, in marked contrast to (rational, real, or complex) numbers whose product is independent of the order of the factors. An example of two matrices not commuting with each other is:

[ ][ ] [ ] 1 2 0 1 0 1 = , 3 4 0 0 0 3 whereas

[ ][ ] [ ] 0 1 1 2 3 4 = . 0 0 3 4 0 0

Besides the ordinary matrix multiplication just described, there exist other less frequently used operations on matrices that can be considered forms of multiplication, such as the Hadamard product and the Kronecker product.[15] They arise in solving matrix equations such as the Sylvester equation.

26.3.3 Row operations

Main article: Row operations

There are three types of row operations:

1. row addition, that is adding a row to another. 2. row multiplication, that is multiplying all entries of a row by a non-zero constant; 3. row switching, that is interchanging two rows of a matrix;

These operations are used in a number of ways, including solving linear equations and finding matrix inverses.

26.3.4 Submatrix

A submatrix of a matrix is obtained by deleting any collection of rows and/or columns.[16][17][18] For example, from the following 3-by-4 matrix, we can construct a 2-by-3 submatrix by removing row 3 and column 2:

  1 2 3 4 [ ] 1 3 4 A = 5 6 7 8  → . 5 7 8 9 10 11 12

The minors and cofactors of a matrix are found by computing the determinant of certain submatrices.[18][19] A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain.[20][21] Other authors define a principal submatrix to be one in which the first k rows and columns, for some number k, are the ones that remain;[22] this type of submatrix has also been called a leading principal submatrix.[23]

26.4 Linear equations

Main articles: Linear equation and System of linear equations

Matrices can be used to compactly write and work with multiple linear equations, i.e., systems of linear equations. For example, if A is an m-by-n matrix, x designates a column vector (i.e., n×1-matrix) of n variables x1, x2, ..., xn, and b is an m×1-column vector, then the matrix equation 162 CHAPTER 26. MATRIX (MATHEMATICS)

Ax = b

is equivalent to the system of linear equations

A₁,₁x1 + A₁,₂x2 + ... + A₁,nxn = b1 ... [24] Am,₁x1 + Am,₂x2 + ... + Am,nxn = bm .

26.5 Linear transformations

Main articles: Linear transformation and Transformation matrix Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. A real m-by-n matrix A gives rise to a linear transformation Rn → Rm mapping each vector x in Rn to the (matrix) product Ax, which is a vector in Rm. Conversely, each linear transformation f: Rn → Rm arises from a unique m-by-n matrix A: explicitly, the (i, j)-entry of A is the ith coordinate of f(ej), where ej = (0,...,0,1,0,...,0) is the unit vector with 1 in the jth position and 0 elsewhere. The matrix A is said to represent the linear map f, and A is called the transformation matrix of f. For example, the 2×2 matrix

[ ] a c A = b d can be viewed as the transform of the unit square into a parallelogram with vertices at (0, 0), (a, b), (a + c, b + d), and[ ] ([c, ]d).[ The] parallelogram[ ] pictured at the right is obtained by multiplying A with each of the column vectors 0 1 1 0 , , and in turn. These vectors define the vertices of the unit square. 0 0 1 1 The following table shows a number of 2-by-2 matrices with the associated linear maps of R2. The blue original is mapped to the green grid and shapes. The origin (0,0) is marked with a black point. Under the 1-to-1 correspondence between matrices and linear maps, matrix multiplication corresponds to composition of maps:[25] if a k-by-m matrix B represents another linear map g : Rm → Rk, then the composition g ∘ f is represented by BA since

(g ∘ f)(x) = g(f(x)) = g(Ax) = B(Ax) = (BA)x.

The last equality follows from the above-mentioned associativity of matrix multiplication. The rank of a matrix A is the maximum number of linearly independent row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors.[26] Equivalently it is the dimension of the image of the linear map represented by A.[27] The rank-nullity theorem states that the dimension of the kernel of a matrix plus the rank equals the number of columns of the matrix.[28]

26.6 Square matrices

Main article: Square matrix

A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. The entries aii form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix.

26.6.1 Main types 26.6. SQUARE MATRICES 163

(a+c,b+d)

(c,d)

ad−bc

(a,b)

(0,0)

The vectors represented by a 2-by-2 matrix correspond to the sides of a unit square transformed into a parallelogram.

Diagonal and triangular matrices

If all entries of A below the main diagonal are zero, A is called an upper triangular matrix. Similarly if all entries of A above the main diagonal are zero, A is called a lower triangular matrix. If all entries outside the main diagonal are zero, A is called a diagonal matrix.

Identity matrix

The identity matrix In of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g. 164 CHAPTER 26. MATRIX (MATHEMATICS)

  1 0 ··· 0 [ ]   [ ] 0 1 ··· 0 1 0 ··· I1 = 1 ,I2 = , ,In = . . . . 0 1 . . .. . 0 0 ··· 1

It is a square matrix of order n, and also a special kind of diagonal matrix. It is called an identity matrix because multiplication with it leaves a matrix unchanged:

AIn = ImA = A for any m-by-n matrix A.

Symmetric or skew-symmetric matrix

A square matrix A that is equal to its transpose, i.e., A = AT, is a symmetric matrix. If instead, A was equal to the negative of its transpose, i.e., A = −AT, then A is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, i.e., the transpose of the complex conjugate of A. By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; i.e., every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.[29] This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see below.

Invertible matrix and its inverse

A square matrix A is called invertible or non-singular if there exists a matrix B such that

AB = BA = In.[30][31]

If B exists, it is unique and is called the inverse matrix of A, denoted A−1.

Definite matrix

A symmetric n×n-matrix is called positive-definite (respectively negative-definite; indefinite), if for all nonzero vectors x ∈ Rn the associated quadratic form given by

Q(x) = xTAx

takes only positive values (respectively only negative values; both some negative and some positive values).[32] If the quadratic form takes only non-negative (respectively only non-positive) values, the symmetric matrix is called positive-semidefinite (respectively negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite. A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, i.e., the matrix is positive- semidefinite and it is invertible.[33] The table at the right shows two possibilities for 2-by-2 matrices. Allowing as input two different vectors instead yields the bilinear form associated to A:

BA (x, y) = xTAy.[34]

Orthogonal matrix

An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors). Equivalently, a matrix A is orthogonal if its transpose is equal to its inverse: 26.6. SQUARE MATRICES 165

AT = A−1, which entails

ATA = AAT = I, where I is the identity matrix. An orthogonal matrix A is necessarily invertible (with inverse A−1 = AT), unitary (A−1 = A*), and normal (A*A = AA*). The determinant of any orthogonal matrix is either +1 or −1. A special orthogonal matrix is an orthogonal matrix with determinant +1. As a linear transformation, every orthogonal matrix with determinant +1 is a pure rotation, while every orthogonal matrix with determinant −1 is either a pure reflection, or a composition of reflection and rotation. The complex analogue of an orthogonal matrix is a unitary matrix.

26.6.2 Main operations

Trace

The trace, tr(A) of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative as mentioned above, the trace of the product of two matrices is independent of the order of the factors:

tr(AB) = tr(BA).

This is immediate from the definition of matrix multiplication:

∑ ∑ m n tr(AB)= i=1 j=1 Aij Bji=tr(BA).

Also, the trace of a matrix is equal to that of its transpose, i.e.,

tr(A) = tr(AT).

Determinant

Main article: Determinant The determinant det(A) or |A| of a square matrix A is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area (in R2) or volume (in R3) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved. The determinant of 2-by-2 matrices is given by

[ ] a b det = ad − bc. c d

The determinant of 3-by-3 matrices involves 6 terms (rule of Sarrus). The more lengthy Leibniz formula generalises these two formulae to all dimensions.[35] The determinant of a product of square matrices equals the product of their determinants:

det(AB) = det(A) · det(B).[36] 166 CHAPTER 26. MATRIX (MATHEMATICS)

0 1 ()1 −1

x2 f(x1 )

x1

f(x2 )

A linear transformation on R2 given by the indicated matrix. The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the orientation, since it turns the counterclockwise orientation of the vectors to a clockwise one.

Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1.[37] Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the Laplace expansion expresses the determinant in terms of minors, i.e., determinants of smaller matrices.[38] This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve linear systems using Cramer’s rule, where the division of the determinants of two related square matrices equates to the value of each of the system’s variables.[39]

Eigenvalues and eigenvectors

Main article: Eigenvalues and eigenvectors

A number λ and a non-zero vector v satisfying

Av = λv

are called an eigenvalue and an eigenvector of A, respectively.[nb 1][40] The number λ is an eigenvalue of an n×n-matrix A if and only if A−λIn is not invertible, which is equivalent to

det(A − λI) = 0. [41]

The polynomial pA in an indeterminate X given by evaluation the determinant det(XIn−A) is called the characteristic polynomial of A. It is a monic polynomial of degree n. Therefore the polynomial equation pA(λ) = 0 has at most n different solutions, i.e., eigenvalues of the matrix.[42] They may be complex even if the entries of A are real. According to the Cayley–Hamilton theorem, pA(A) = 0, that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix.

26.7 Computational aspects

Matrix calculations can be often performed with different techniques. Many problems can be solved by both direct algorithms or iterative approaches. For example, the eigenvectors of a square matrix can be obtained by finding a sequence of vectors xn converging to an eigenvector when n tends to infinity.[43] 26.8. DECOMPOSITION 167

To be able to choose the more appropriate algorithm for each specific problem, it is important to determine both the effectiveness and precision of all the available algorithms. The domain studying these matters is called numerical linear algebra.[44] As with other numerical situations, two main aspects are the complexity of algorithms and their numerical stability. Determining the complexity of an algorithm means finding upper bounds or estimates of how many elementary oper- ations such as additions and multiplications of scalars are necessary to perform some algorithm, e.g., multiplication of matrices. For example, calculating the matrix product of two n-by-n matrix using the definition given above needs n3 multiplications, since for any of the n2 entries of the product, n multiplications are necessary. The Strassen algo- rithm outperforms this “naive” algorithm; it needs only n2.807 multiplications.[45] A refined approach also incorporates specific features of the computing devices. In many practical situations additional information about the matrices involved is known. An important case are sparse matrices, i.e., matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the conjugate gradient method.[46] An algorithm is, roughly speaking, numerically stable, if little deviations in the input values do not lead to big de- viations in the result. For example, calculating the inverse of a matrix via Laplace’s formula (Adj (A) denotes the adjugate matrix of A)

A−1 = Adj(A) / det(A)

may lead to significant rounding errors if the determinant of the matrix is very small. The norm of a matrix can be used to capture the conditioning of linear algebraic problems, such as computing a matrix’s inverse.[47] Although most computer languages are not designed with commands or libraries for matrices, as early as the 1970s, some engineering desktop computers such as the HP 9830 had ROM cartridges to add BASIC commands for matrices. Some computer languages such as APL were designed to manipulate matrices, and various mathematical programs can be used to aid computing with matrices.[48]

26.8 Decomposition

Main articles: Matrix decomposition, Matrix diagonalization, Gaussian elimination and Montante’s method

There are several methods to render matrices into a more easily accessible form. They are generally referred to as matrix decomposition or matrix factorization techniques. The interest of all these techniques is that they preserve certain properties of the matrices in question, such as determinant, rank or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices. The LU decomposition factors matrices as a product of lower (L) and an upper triangular matrices (U).[49] Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The Gaus- sian elimination is a similar algorithm; it transforms any matrix to row echelon form.[50] Both methods proceed by multiplying the matrix by suitable elementary matrices, which correspond to permuting rows or columns and adding multiples of one row to another row. Singular value decomposition expresses any matrix A as a product UDV∗, where U and V are unitary matrices and D is a diagonal matrix. The eigendecomposition or diagonalization expresses A as a product VDV−1, where D is a diagonal matrix and V is a suitable invertible matrix.[51] If A can be written in this form, it is called diagonalizable. More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into Jordan normal form, that is to say matrices whose only nonzero entries are the eigenvalues λ1 to λ of A, placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right.[52] Given the eigendecomposition, the nth power of A (i.e., n-fold iterated matrix multiplication) can be calculated via

An = (VDV−1)n = VDV−1VDV−1...VDV−1 = VDnV−1

and the power of a diagonal matrix can be calculated by taking the corresponding powers of the diagonal entries, which is much easier than doing the exponentiation for A instead. This can be used to compute the matrix exponential eA, a 168 CHAPTER 26. MATRIX (MATHEMATICS)

An example of a matrix in Jordan normal form. The grey blocks are called Jordan blocks. need frequently arising in solving linear differential equations, matrix logarithms and square roots of matrices.[53] To avoid numerically ill-conditioned situations, further algorithms such as the Schur decomposition can be employed.[54]

26.9 Abstract algebraic aspects and generalizations

Matrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general fields or even rings, while linear algebra codifies properties of matrices in the notion of linear maps. It is possible to consider matrices with infinitely many columns and rows. Another extension are tensors, which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realised as sequences of numbers, while matrices are rectangular or two-dimensional arrays of numbers.[55] Matrices, subject to certain requirements tend to form groups known as matrix groups.

26.9.1 Matrices with more general entries

This article focuses on matrices whose entries are real or complex numbers. However, matrices can be considered with much more general types of entries than real or complex numbers. As a first step of generalization, any field, i.e., a set where addition, subtraction, multiplication and division operations are defined and well-behaved, may be used instead of R or C, for example rational numbers or finite fields. For example, coding theory makes use of matrices over finite fields. Wherever eigenvalues are considered, as these are roots of a polynomial they may exist only in a 26.9. ABSTRACT ALGEBRAIC ASPECTS AND GENERALIZATIONS 169 larger field than that of the entries of the matrix; for instance they may be complex in case of a matrix with real entries. The possibility to reinterpret the entries of a matrix as elements of a larger field (e.g., to view a real matrix as a complex matrix whose entries happen to be all real) then allows considering each square matrix to possess a full set of eigenvalues. Alternatively one can consider only matrices with entries in an algebraically closed field, such as C, from the outset. More generally, abstract algebra makes great use of matrices with entries in a ring R.[56] Rings are a more general notion than fields in that a division operation need not exist. The very same addition and multiplication operations of matrices extend to this setting, too. The set M(n, R) of all square n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R-module Rn.[57] If the ring R is commutative, i.e., its multiplication is commutative, then M(n, R) is a unitary noncommutative (unless n = 1) associative algebra over R. The determinant of square matrices over a commutative ring R can still be defined using the Leibniz formula; such a matrix is invertible if and only if its determinant is invertible in R, generalising the situation over a field F, where every nonzero element is invertible.[58] Matrices over superrings are called supermatrices.[59] Matrices do not always have all their entries in the same ring – or even in any ring at all. One special but common case is block matrices, which may be considered as matrices whose entries themselves are matrices. The entries need not be quadratic matrices, and thus need not be members of any ordinary ring; but their sizes must fulfil certain compatibility conditions.

26.9.2 Relationship to linear maps

Linear maps Rn → Rm are equivalent to m-by-n matrices, as described above. More generally, any linear map f: V → W between finite-dimensional vector spaces can be described by a matrix A = (aij), after choosing bases v1, ..., vn of V, and w1, ..., wm of W (so n is the dimension of V and m is the dimension of W), which is such that

∑m f(vj) = ai,jwi for j = 1, . . . , n. i=1 In other words, column j of A expresses the image of vj in terms of the basis vectors wi of W; thus this relation uniquely determines the entries of the matrix A. Note that the matrix depends on the choice of the bases: different choices of bases give rise to different, but equivalent matrices.[60] Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix AT describes the transpose of the linear map given by A, with respect to the dual bases.[61] These properties can be restated in a more natural way: the category of all matrices with entries in a field k with multiplication as composition is equivalent to the category of finite dimensional vector spaces and linear maps over this field. More generally, the set of m×n matrices can be used to represent the R-linear maps between the free modules Rm and Rn for an arbitrary ring R with unity. When n = m composition of these maps is possible, and this gives rise to the matrix ring of n×n matrices representing the endomorphism ring of Rn.

26.9.3 Matrix groups

Main article: Matrix group

A group is a mathematical structure consisting of a set of objects together with a binary operation, i.e., an operation combining any two objects to a third, subject to certain requirements.[62] A group in which the objects are matrices and the group operation is matrix multiplication is called a matrix group.[nb 2][63] Since in a group every element has to be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the general linear groups. Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups. For example, matrices with a given size and with a determinant of 1 form a subgroup of (i.e., a smaller group contained in) their general linear group, called a special linear group.[64] Orthogonal matrices, determined by the condition

MTM = I, 170 CHAPTER 26. MATRIX (MATHEMATICS)

form the orthogonal group.[65] Every orthogonal matrix has determinant 1 or −1. Orthogonal matrices with determi- nant 1 form a subgroup called special orthogonal group. Every finite group is isomorphic to a matrix group, as one can see by considering the regular representation of the symmetric group.[66] General groups can be studied using matrix groups, which are comparatively well-understood, by means of representation theory.[67]

26.9.4 Infinite matrices

It is also possible to consider matrices with infinitely many rows and/or columns[68] even if, being infinite objects, one cannot write down such matrices explicitly. All that matters is that for every element in the set indexing rows, and every element in the set indexing columns, there is a well-defined entry (these index sets need not even be subsets of the natural numbers). The basic operations of addition, subtraction, scalar multiplication and transposition can still be defined without problem; however matrix multiplication may involve infinite summations to define the resulting entries, and these are not defined in general. ⊕ If R is any ring with unity, then the ring of endomorphisms of M = i∈I R as a right R module is isomorphic to the ring of column finite matrices CFMI (R) whose entries are indexed by I × I , and whose columns each contain only finitely many nonzero entries. The endomorphisms of M considered as a left R module result in an analogous object, the row finite matrices RFMI (R) whose rows each only have finitely many nonzero entries. If infinite matrices are used to describe linear maps, then only those matrices can be used all of whose columns have but a finite number of nonzero entries, for the following reason. For a matrix A to describe a linear map f: V→W, bases for both spaces must have been chosen; recall that by definition this means that every vector in the space can be written uniquely as a (finite) linear combination of basis vectors, so that written as a (column) vector v of coefficients, only finitely many entries vi are nonzero. Now the columns of A describe the images by f of individual basis vectors of V in the basis of W, which is only meaningful if these columns have only finitely many nonzero entries. There is no restriction on the rows of A however: in the product A·v there are only finitely many nonzero coefficients of v involved, so every one of its entries, even if it is given as an infinite sum of products, involves only finitely many nonzero terms and is therefore well defined. Moreover, this amounts to forming a linear combination of the columns of A that effectively involves only finitely many of them, whence the result has only finitely many nonzero entries, because each of those columns do. One also sees that products of two matrices of the given type is well defined (provided as usual that the column-index and row-index sets match), is again of the same type, and corresponds to the composition of linear maps. If R is a normed ring, then the condition of row or column finiteness can be relaxed. With the norm in place, absolutely convergent series can be used instead of finite sums. For example, the matrices whose column sums are absolutely convergent sequences form a ring. Analogously of course, the matrices whose row sums are absolutely convergent series also form a ring. In that vein, infinite matrices can also be used to describe operators on Hilbert spaces, where convergence and continuity questions arise, which again results in certain constraints that have to be imposed. However, the explicit point of view of matrices tends to obfuscate the matter,[nb 3] and the abstract and more powerful tools of functional analysis can be used instead.

26.9.5 Empty matrices

An empty matrix is a matrix in which the number of rows or columns (or both) is zero.[69][70] Empty matrices help dealing with maps involving the zero vector space. For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them. The determinant of the 0-by-0 matrix is 1 as follows from regarding the empty product occurring in the Leibniz formula for the determinant as 1. This value is also consistent with the fact that the identity map from any finite dimensional space to itself has determinant 1, a fact that is often used as a part of the characterization of determinants. 26.10. APPLICATIONS 171

26.10 Applications

There are numerous applications of matrices, both in mathematics and other sciences. Some of them merely take advantage of the compact representation of a set of numbers in a matrix. For example, in game theory and economics, the payoff matrix encodes the payoff for two players, depending on which out of a given (finite) set of alternatives the players choose.[71] Text mining and automated thesaurus compilation makes use of document-term matrices such as tf-idf to track frequencies of certain words in several documents.[72] Complex numbers can be represented by particular real 2-by-2 matrices via

[ ] a −b a + ib ↔ , b a

under which addition and multiplication of complex numbers and matrices correspond to each other. For example, 2-by-2 rotation matrices represent the multiplication with some complex number of absolute value 1, as above.A similar interpretation is possible for quaternions[73] and Clifford algebras in general. Early encryption techniques such as the Hill cipher also used matrices. However, due to the linear nature of matrices, these codes are comparatively easy to break.[74] Computer graphics uses matrices both to represent objects and to calculate transformations of objects using affine rotation matrices to accomplish tasks such as projecting a three- dimensional object onto a two-dimensional screen, corresponding to a theoretical camera observation.[75] Matrices over a polynomial ring are important in the study of control theory. Chemistry makes use of matrices in various ways, particularly since the use of quantum theory to discuss molecular bonding and spectroscopy. Examples are the overlap matrix and the Fock matrix used in solving the Roothaan equations to obtain the molecular orbitals of the Hartree–Fock method.

26.10.1 Graph theory

The adjacency matrix of a finite graph is a basic notion of graph theory.[76] It records which vertices of the graph are connected by an edge. Matrices containing just two different values (1 and 0 meaning for example “yes” and “no”, respectively) are called logical matrices. The distance (or cost) matrix contains information about distances of the edges.[77] These concepts can be applied to websites connected by hyperlinks or cities connected by roads etc., in which case (unless the connection network is extremely dense) the matrices tend to be sparse, i.e., contain few nonzero entries. Therefore, specifically tailored matrix algorithms can be used in network theory.

26.10.2 Analysis and geometry

The Hessian matrix of a differentiable function ƒ: Rn → R consists of the second derivatives of ƒ with respect to the several coordinate directions, i.e.[78]

[ ] ∂2f H(f) = . ∂xi ∂xj

It encodes information about the local growth behaviour of the function: given a critical point x = (x1, ..., xn), i.e., a point where the first partial derivatives ∂f/∂xi of ƒ vanish, the function has a local minimum if the Hessian matrix is positive definite. Quadratic programming can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see above).[79] Another matrix frequently used in geometrical situations is the Jacobi matrix of a differentiable map f: Rn → Rm. If [80] f1, ..., fm denote the components of f, then the Jacobi matrix is defined as

[ ] ∂f J(f) = i . ∂xj 1≤i≤m,1≤j≤n If n > m, and if the rank of the Jacobi matrix attains its maximal value m, f is locally invertible at that point, by the implicit function theorem.[81] 172 CHAPTER 26. MATRIX (MATHEMATICS)

2 3

1

  1 1 0 An undirected graph with adjacency matrix 1 0 1. 0 1 0

Partial differential equations can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation. For elliptic partial differential equations this matrix is positive definite, which has decisive influence on the set of possible solutions of the equation in question.[82] The finite element method is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems. It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen with respect to a sufficiently fine grid, which in turn can be recast as a matrix equation.[83]

26.10.3 Probability theory and statistics

Stochastic matrices are square matrices whose rows are probability vectors, i.e., whose entries are non-negative and sum up to one. Stochastic matrices are used to define Markov chains with finitely many states.[84] A row of the stochastic matrix gives the probability distribution for the next position of some particle currently in the state that corresponds to the row. Properties of the Markov chain like absorbing states, i.e., states that any particle attains eventually, can be read off the eigenvectors of the transition matrices.[85] Statistics also makes use of matrices in many different forms.[86] Descriptive statistics is concerned with describing data sets, which can often be represented as data matrices, which may then be subjected to dimensionality reduction techniques. The covariance matrix encodes the mutual variance of several random variables.[87] Another technique 26.10. APPLICATIONS 173

[ ] 2 0 At the saddle point (x = 0, y = 0) (red) of the function f(x,−y) = x2 − y2, the Hessian matrix is indefinite. 0 −2

using matrices are linear least squares, a method that approximates a finite set of pairs (x1, y1), (x2, y2), ..., (xN, yN), by a linear function

yi ≈ axi + b, i = 1, ..., N

which can be formulated in terms of matrices, related to the singular value decomposition of matrices.[88] Random matrices are matrices whose entries are random numbers, subject to suitable probability distributions, such as matrix normal distribution. Beyond probability theory, they are applied in domains ranging from number theory to physics.[89][90]

26.10.4 Symmetries and transformations in physics

Further information: Symmetry in physics

Linear transformations and the associated symmetries play a key role in modern physics. For example, elementary particles in quantum field theory are classified as representations of the Lorentz group of special relativity and, more specifically, by their behavior under the spin group. Concrete representations involving the Pauli matrices and more general gamma matrices are an integral part of the physical description of fermions, which behave as spinors.[91] For the three lightest quarks, there is a group-theoretical representation involving the special unitary group SU(3); for their calculations, physicists use a convenient matrix representation known as the Gell-Mann matrices, which are also used for the SU(3) gauge group that forms the basis of the modern description of strong nuclear interactions, quantum chromodynamics. The Cabibbo–Kobayashi–Maskawa matrix, in turn, expresses the fact that the basic quark states that are important for weak interactions are not the same as, but linearly related to the basic quark states that define particles with specific and distinct masses.[92] 174 CHAPTER 26. MATRIX (MATHEMATICS)

Two different Markov chains. The chart depicts the number of[ particles] (of a total[ of 1000)] in state “2”. Both limiting values can be .7 0 .7 .2 determined from the transition matrices, which are given by (red) and (black). .3 1 .3 .8

26.10.5 Linear combinations of quantum states

The first model of quantum mechanics (Heisenberg, 1925) represented the theory’s operators by infinite-dimensional matrices acting on quantum states.[93] This is also referred to as matrix mechanics. One particular example is the density matrix that characterizes the “mixed” state of a quantum system as a linear combination of elementary, “pure” eigenstates.[94] Another matrix serves as a key tool for describing the scattering experiments that form the cornerstone of experimen- tal particle physics: Collision reactions such as occur in particle accelerators, where non-interacting particles head towards each other and collide in a small interaction zone, with a new set of non-interacting particles as the result, can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states. The linear combination is given by a matrix known as the S-matrix, which encodes all information about the possible interactions between particles.[95]

26.10.6 Normal modes

A general application of matrices in physics is to the description of linearly coupled harmonic systems. The equations of motion of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a force matrix multiplying a displacement vector to characterize the interactions. The best way to obtain solutions is to determine the system’s eigenvectors, its normal modes, by diagonalizing the matrix equation. Techniques like this are crucial when it comes to the internal dynamics of molecules: the internal vibra- tions of systems consisting of mutually bound component atoms.[96] They are also needed for describing mechanical vibrations, and oscillations in electrical circuits.[97] 26.11. HISTORY 175

26.10.7 Geometrical optics

Geometrical optics provides further matrix applications. In this approximative theory, the wave nature of light is neglected. The result is a model in which light rays are indeed geometrical rays. If the deflection of light rays by optical elements is small, the action of a lens or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called ray transfer matrix: the vector’s components are the light ray’s slope and its distance from the optical axis, while the matrix encodes the properties of the optical element. Actually, there are two kinds of matrices, viz. a refraction matrix describing the refraction at a lens surface, and a translation matrix, describing the translation of the plane of reference to the next refracting surface, where another refraction matrix applies. The optical system, consisting of a combination of lenses and/or reflective elements, is simply described by the matrix resulting from the product of the components’ matrices.[98]

26.10.8 Electronics

Traditional mesh analysis in electronics leads to a system of linear equations that can be described with a matrix. The behaviour of many electronic components can be described using matrices. Let A be a 2-dimensional vector with the component’s input voltage v1 and input current i1 as its elements, and let B be a 2-dimensional vector with the component’s output voltage v2 and output current i2 as its elements. Then the behaviour of the electronic component can be described by B = H · A, where H is a 2 x 2 matrix containing one impedance element (h12), one admittance element (h21) and two dimensionless elements (h11 and h22). Calculating a circuit now reduces to multiplying matrices.

26.11 History

Matrices have a long history of application in solving linear equations but they were known as arrays until the 1800s. The Chinese text The Nine Chapters on the Mathematical Art written in 10th–2nd century BCE is the first example of the use of array methods to solve simultaneous equations,[99] including the concept of determinants. In 1545 Italian mathematician Girolamo Cardano brought the method to Europe when he published Ars Magna.[100] The Japanese mathematician Seki used the same array methods to solve simultaneous equations in 1683.[101] The Dutch Mathematician Jan de Witt represented transformations using arrays in his 1659 book Elements of Curves (1659).[102] Between 1700 and 1710 Gottfried Wilhelm Leibniz publicized the use of arrays for recording information or solutions and experimented with over 50 different systems of arrays.[100] Cramer presented his rule in 1750. The term “matrix” (Latin for “womb”, derived from mater—mother[103]) was coined by James Joseph Sylvester in 1850,[104] who understood a matrix as an object giving rise to a number of determinants today called minors, that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows. In an 1851 paper, Sylvester explains:

I have in previous papers defined a “Matrix” as a rectangular array of terms, out of which different systems of determinants may be engendered as from the womb of a common parent.[105]

Arthur Cayley published a treatise on geometric transformations using matrices that were not rotated versions of the coefficients being investigated as had previously been done. Instead he defined operations such as addition, subtrac- tion, multiplication, and division as transformations of those matrices and showed the associative and distributive properties held true. Cayley investigated and demonstrated the non-commutative property of matrix multiplication as well as the commutative property of matrix addition.[100] Early matrix theory had limited the use of arrays almost exclusively to determinants and Arthur Cayley’s abstract matrix operations were revolutionary. He was instrumental in proposing a matrix concept independent of equation systems. In 1858 Cayley published his Memoir on the theory of matrices[106][107] in which he proposed and demonstrated the Cayley-Hamilton theorem.[100] An English mathematician named Cullis was the first to use modern bracket notation for matrices in 1913 and he simultaneously demonstrated the first significant use of the notation A = [ai,j] to represent a matrix where ai,j refers to the ith row and the jth column.[100] The study of determinants sprang from several sources.[108] Number-theoretical problems led Gauss to relate coef- ficients of quadratic forms, i.e., expressions such as x2 + xy − 2y2, and linear maps in three dimensions to matri- ces. Eisenstein further developed these notions, including the remark that, in modern parlance, matrix products are 176 CHAPTER 26. MATRIX (MATHEMATICS)

non-commutative. Cauchy was the first to prove general statements about determinants, using as definition of the determinant of a matrix A = [ai,j] the following: replace the powers ajk by ajk in the polynomial

∏ a1a2 ··· an (aj − ai) ihypercomplex number systems of the previous century. The inception of matrix mechanics by Heisenberg, Born and Jordan led to studying matrices with infinitely many rows and columns.[113] Later, von Neumann carried out the mathematical formulation of quantum mechanics, by further developing functional analytic notions such as linear operators on Hilbert spaces, which, very roughly speaking, correspond to Euclidean space, but with an infinity of independent directions.

26.11.1 Other historical usages of the word “matrix” in mathematics

The word has been used in unusual ways by at least two authors of historical importance. Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910–1913) use the word “matrix” in the context of their Axiom of reducibility. They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the “bottom” (0 order) the function is identical to its extension:

“Let us give the name of matrix to any function, of however many variables, which does not involve any apparent variables. Then any possible function other than a matrix is derived from a matrix by means of generalization, i.e., by considering the proposition which asserts that the function in question is true with all possible values or with some value of one of the arguments, the other argument or arguments remaining undetermined”.[114]

For example, a function Φ(x, y) of two variables x and y can be reduced to a collection of functions of a single variable, e.g., y, by “considering” the function for all possible values of “individuals” ai substituted in place of variable x. And then the resulting collection of functions of the single variable y, i.e., ∀aᵢ: Φ(ai, y), can be reduced to a “matrix” of values by “considering” the function for all possible values of “individuals” bi substituted in place of variable y:

∀b∀aᵢ: Φ(ai, b).

Alfred Tarski in his 1946 Introduction to Logic used the word “matrix” synonymously with the notion of truth table as used in mathematical logic.[115]

26.12 See also

• Algebraic multiplicity • Geometric multiplicity • Gram-Schmidt process • List of matrices 26.13. NOTES 177

• Matrix calculus • Periodic matrix set • Tensor

26.13 Notes

[1] equivalently, table

[2] Anton (1987, p. 23)

[3] Beauregard & Fraleigh (1973, p. 56)

[4] Young, Cynthia. Precalculus. Laurie Rosatone. p. 727. Check date values in: |access-date= (help);

[5] K. Bryan and T. Leise. The $25,000,000,000 eigenvector: The linear algebra behind Google. SIAM Review, 48(3):569– 581, 2006.

[6] Lang 2002

[7] Fraleigh (1976, p. 209)

[8] Nering (1970, p. 37)

[9] Oualline 2003, Ch. 5

[10] “How to organize, add and multiply matrices - Bill Shillito”. TED ED. Retrieved April 6, 2013.

[11] Brown 1991, Definition I.2.1 (addition), Definition I.2.4 (scalar multiplication), and Definition I.2.33 (transpose)

[12] Brown 1991, Theorem I.2.6

[13] Brown 1991, Definition I.2.20

[14] Brown 1991, Theorem I.2.24

[15] Horn & Johnson 1985, Ch. 4 and 5

[16] Bronson (1970, p. 16)

[17] Kreyszig (1972, p. 220)

[18] Protter & Morrey (1970, p. 869)

[19] Kreyszig (1972, pp. 241,244)

[20] Schneider, Hans; Barker, George Phillip (2012), Matrices and Linear Algebra, Dover Books on Mathematics, Courier Dover Corporation, p. 251, ISBN 9780486139302.

[21] Perlis, Sam (1991), Theory of Matrices, Dover books on advanced mathematics, Courier Dover Corporation, p. 103, ISBN 9780486668109.

[22] Anton, Howard (2010), Elementary Linear Algebra (10th ed.), John Wiley & Sons, p. 414, ISBN 9780470458211.

[23] Horn, Roger A.; Johnson, Charles R. (2012), Matrix Analysis (2nd ed.), Cambridge University Press, p. 17, ISBN 9780521839402.

[24] Brown 1991, I.2.21 and 22

[25] Greub 1975, Section III.2

[26] Brown 1991, Definition II.3.3

[27] Greub 1975, Section III.1

[28] Brown 1991, Theorem II.3.22

[29] Horn & Johnson 1985, Theorem 2.5.6

[30] Brown 1991, Definition I.2.28 178 CHAPTER 26. MATRIX (MATHEMATICS)

[31] Brown 1991, Definition I.5.13

[32] Horn & Johnson 1985, Chapter 7

[33] Horn & Johnson 1985, Theorem 7.2.1

[34] Horn & Johnson 1985, Example 4.0.6, p. 169

[35] Brown 1991, Definition III.2.1

[36] Brown 1991, Theorem III.2.12

[37] Brown 1991, Corollary III.2.16

[38] Mirsky 1990, Theorem 1.4.1

[39] Brown 1991, Theorem III.3.18

[40] Brown 1991, Definition III.4.1

[41] Brown 1991, Definition III.4.9

[42] Brown 1991, Corollary III.4.10

[43] Householder 1975, Ch. 7

[44] Bau III & Trefethen 1997

[45] Golub & Van Loan 1996, Algorithm 1.3.1

[46] Golub & Van Loan 1996, Chapters 9 and 10, esp. section 10.2

[47] Golub & Van Loan 1996, Chapter 2.3

[48] For example, Mathematica, see Wolfram 2003, Ch. 3.7

[49] Press, Flannery & Teukolsky 1992

[50] Stoer & Bulirsch 2002, Section 4.1

[51] Horn & Johnson 1985, Theorem 2.5.4

[52] Horn & Johnson 1985, Ch. 3.1, 3.2

[53] Arnold & Cooke 1992, Sections 14.5, 7, 8

[54] Bronson 1989, Ch. 15

[55] Coburn 1955, Ch. V

[56] Lang 2002, Chapter XIII

[57] Lang 2002, XVII.1, p. 643

[58] Lang 2002, Proposition XIII.4.16

[59] Reichl 2004, Section L.2

[60] Greub 1975, Section III.3

[61] Greub 1975, Section III.3.13

[62] See any standard reference in group.

[63] Baker 2003, Def. 1.30

[64] Baker 2003, Theorem 1.2

[65] Artin 1991, Chapter 4.5

[66] Rowen 2008, Example 19.2, p. 198

[67] See any reference in representation theory or group representation.

[68] See the item “Matrix” in Itõ, ed. 1987 26.13. NOTES 179

[69] “Empty Matrix: A matrix is empty if either its row or column dimension is zero”, Glossary, O-Matrix v6 User Guide

[70] “A matrix having at least one dimension equal to zero is called an empty matrix”, MATLAB Data Structures

[71] Fudenberg & Tirole 1983, Section 1.1.1

[72] Manning 1999, Section 15.3.4

[73] Ward 1997, Ch. 2.8

[74] Stinson 2005, Ch. 1.1.5 and 1.2.4

[75] Association for Computing Machinery 1979, Ch. 7

[76] Godsil & Royle 2004, Ch. 8.1

[77] Punnen 2002

[78] Lang 1987a, Ch. XVI.6

[79] Nocedal 2006, Ch. 16

[80] Lang 1987a, Ch. XVI.1

[81] Lang 1987a, Ch. XVI.5. For a more advanced, and more general statement see Lang 1969, Ch. VI.2

[82] Gilbarg & Trudinger 2001

[83] Šolin 2005, Ch. 2.5. See also stiffness method.

[84] Latouche & Ramaswami 1999

[85] Mehata & Srinivasan 1978, Ch. 2.8

[86] Healy, Michael (1986), Matrices for Statistics, Oxford University Press, ISBN 978-0-19-850702-4

[87] Krzanowski 1988, Ch. 2.2., p. 60

[88] Krzanowski 1988, Ch. 4.1

[89] Conrey 2007

[90] Zabrodin, Brezin & Kazakov et al. 2006

[91] Itzykson & Zuber 1980, Ch. 2

[92] see Burgess & Moore 2007, section 1.6.3. (SU(3)), section 2.4.3.2. (Kobayashi–Maskawa matrix)

[93] Schiff 1968, Ch. 6

[94] Bohm 2001, sections II.4 and II.8

[95] Weinberg 1995, Ch. 3

[96] Wherrett 1987, part II

[97] Riley, Hobson & Bence 1997, 7.17

[98] Guenther 1990, Ch. 5

[99] Shen, Crossley & Lun 1999 cited by Bretscher 2005, p. 1

[100] Discrete Mathematics 4th Ed. Dossey, Otto, Spense, Vanden Eynden, Published by Addison Wesley, October 10, 2001 ISBN 978-0321079121 | p.564-565

[101] Needham, Joseph; Wang Ling (1959). Science and Civilisation in China III. Cambridge: Cambridge University Press. p. 117. ISBN 9780521058018.

[102] Discrete Mathematics 4th Ed. Dossey, Otto, Spense, Vanden Eynden, Published by Addison Wesley, October 10, 2001 ISBN 978-0321079121 | p.564

[103] Merriam–Webster dictionary, Merriam–Webster, retrieved April 20, 2009 180 CHAPTER 26. MATRIX (MATHEMATICS)

[104] Although many sources state that J. J. Sylvester coined the mathematical term “matrix” in 1848, Sylvester published nothing in 1848. (For proof that Sylvester published nothing in 1848, see: J. J. Sylvester with H. F. Baker, ed., The Collected Mathematical Papers of James Joseph Sylvester (Cambridge, England: Cambridge University Press, 1904), vol. 1.) His earliest use of the term “matrix” occurs in 1850 in: J. J. Sylvester (1850) “Additions to the articles in the September number of this journal, “On a new class of theorems,” and on Pascal’s theorem,” The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 37 : 363-370. From page 369: “For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants … "

[105] The Collected Mathematical Papers of James Joseph Sylvester: 1837–1853, Paper 37, p. 247

[106] Phil.Trans. 1858, vol.148, pp.17-37 Math. Papers II 475-496

[107] Dieudonné, ed. 1978, Vol. 1, Ch. III, p. 96

[108] Knobloch 1994

[109] Hawkins 1975

[110] Kronecker 1897

[111] Weierstrass 1915, pp. 271–286

[112] Bôcher 2004

[113] Mehra & Rechenberg 1987

[114] Whitehead, Alfred North; and Russell, Bertrand (1913) Principia Mathematica to *56, Cambridge at the University Press, Cambridge UK (republished 1962) cf page 162ff.

[115] Tarski, Alfred; (1946) Introduction to Logic and the Methodology of Deductive Sciences, Dover Publications, Inc, New York NY, ISBN 0-486-28462-X.

[1] Eigen means “own” in German and in Dutch.

[2] Additionally, the group is required to be closed in the general linear group.

[3] “Not much of matrix theory carries over to infinite-dimensional spaces, and what does is not so useful, but it sometimes helps.” Halmos 1982, p. 23, Chapter 5

26.14 References

• Anton, Howard (1987), Elementary Linear Algebra (5th ed.), New York: Wiley, ISBN 0-471-84819-0 • Arnold, Vladimir I.; Cooke, Roger (1992), Ordinary differential equations, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-3-540-54813-3 • Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1 • Association for Computing Machinery (1979), Computer Graphics, Tata McGraw–Hill, ISBN 978-0-07-059376- 3 • Baker, Andrew J. (2003), Matrix Groups: An Introduction to Lie Group Theory, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-1-85233-470-3 • Bau III, David; Trefethen, Lloyd N. (1997), Numerical linear algebra, Philadelphia, PA: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-361-9 • Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduc- tion to Groups, Rings, and Fields, Boston: Houghton Mifflin Co., ISBN 0-395-14017-X • Bretscher, Otto (2005), Linear Algebra with Applications (3rd ed.), Prentice Hall • Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490 • Bronson, Richard (1989), Schaum’s outline of theory and problems of matrix operations, New York: McGraw– Hill, ISBN 978-0-07-007978-6 26.14. REFERENCES 181

• Brown, William C. (1991), Matrices and vector spaces, New York, NY: Marcel Dekker, ISBN 978-0-8247- 8419-5

• Coburn, Nathaniel (1955), Vector and tensor analysis, New York, NY: Macmillan, OCLC 1029828

• Conrey, J. Brian (2007), Ranks of elliptic curves and random matrix theory, Cambridge University Press, ISBN 978-0-521-69964-8

• Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0- 201-01984-1

• Fudenberg, Drew; Tirole, Jean (1983), Game Theory, MIT Press

• Gilbarg, David; Trudinger, Neil S. (2001), Elliptic partial differential equations of second order (2nd ed.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-3-540-41160-4

• Godsil, Chris; Royle, Gordon (2004), Algebraic Graph Theory, Graduate Texts in Mathematics 207, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-95220-8

• Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Johns Hopkins, ISBN 978-0- 8018-5414-9

• Greub, Werner Hildbert (1975), Linear algebra, Graduate Texts in Mathematics, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-90110-7

• Halmos, Paul Richard (1982), A Hilbert space problem book, Graduate Texts in Mathematics 19 (2nd ed.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-90685-0, MR 675952

• Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521- 38632-6

• Householder, Alston S. (1975), The theory of matrices in numerical analysis, New York, NY: Dover Publica- tions, MR 0378371

• Kreyszig, Erwin (1972), Advanced Engineering Mathematics (3rd ed.), New York: Wiley, ISBN 0-471-50728- 8.

• Krzanowski, Wojtek J. (1988), Principles of multivariate analysis, Oxford Statistical Science Series 3, The Clarendon Press Oxford University Press, ISBN 978-0-19-852211-9, MR 969370

• Itõ, Kiyosi, ed. (1987), Encyclopedic dictionary of mathematics. Vol. I-IV (2nd ed.), MIT Press, ISBN 978-0- 262-09026-1, MR 901762

• Lang, Serge (1969), Analysis II, Addison-Wesley

• Lang, Serge (1987a), Calculus of several variables (3rd ed.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-96405-8

• Lang, Serge (1987b), Linear algebra, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-96412-6

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer- Verlag, ISBN 978-0-387-95385-4, MR 1878556

• Latouche, Guy; Ramaswami, Vaidyanathan (1999), Introduction to matrix analytic methods in stochastic mod- eling (1st ed.), Philadelphia, PA: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-425-8

• Manning, Christopher D.; Schütze, Hinrich (1999), Foundations of statistical natural language processing, MIT Press, ISBN 978-0-262-13360-9

• Mehata, K. M.; Srinivasan, S. K. (1978), Stochastic processes, New York, NY: McGraw–Hill, ISBN 978-0-07- 096612-3

• Mirsky, Leonid (1990), An Introduction to Linear Algebra, Courier Dover Publications, ISBN 978-0-486- 66434-7

• Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76-91646 182 CHAPTER 26. MATRIX (MATHEMATICS)

• Nocedal, Jorge; Wright, Stephen J. (2006), Numerical Optimization (2nd ed.), Berlin, DE; New York, NY: Springer-Verlag, p. 449, ISBN 978-0-387-30303-1

• Oualline, Steve (2003), Practical C++ programming, O'Reilly, ISBN 978-0-596-00419-4

• Press, William H.; Flannery, Brian P.; Teukolsky, Saul A.; Vetterling, William T. (1992), “LU Decomposi- tion and Its Applications”, Numerical Recipes in FORTRAN: The Art of Scientific Computing (PDF) (2nd ed.), Cambridge University Press, pp. 34–42

• Protter, Murray H.; Morrey, Jr., Charles B. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042

• Punnen, Abraham P.; Gutin, Gregory (2002), The traveling salesman problem and its variations, Boston, MA: Kluwer Academic Publishers, ISBN 978-1-4020-0664-7

• Reichl, Linda E. (2004), The transition to chaos: conservative classical systems and quantum manifestations, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-98788-0

• Rowen, Louis Halle (2008), Graduate Algebra: noncommutative view, Providence, RI: American Mathematical Society, ISBN 978-0-8218-4153-2

• Šolin, Pavel (2005), Partial Differential Equations and the Finite Element Method, Wiley-Interscience, ISBN 978-0-471-76409-0

• Stinson, Douglas R. (2005), Cryptography, Discrete Mathematics and its Applications, Chapman & Hall/CRC, ISBN 978-1-58488-508-5

• Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd ed.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-95452-3

• Ward, J. P. (1997), Quaternions and Cayley numbers, Mathematics and its Applications 403, Dordrecht, NL: Kluwer Academic Publishers Group, ISBN 978-0-7923-4513-8, MR 1458894

• Wolfram, Stephen (2003), The Mathematica Book (5th ed.), Champaign, IL: Wolfram Media, ISBN 978-1- 57955-022-6

26.14.1 Physics references

• Bohm, Arno (2001), Quantum Mechanics: Foundations and Applications, Springer, ISBN 0-387-95330-2

• Burgess, Cliff; Moore, Guy (2007), The Standard Model. A Primer, Cambridge University Press, ISBN 0-521- 86036-9

• Guenther, Robert D. (1990), Modern Optics, John Wiley, ISBN 0-471-60538-7

• Itzykson, Claude; Zuber, Jean-Bernard (1980), Quantum Field Theory, McGraw–Hill, ISBN 0-07-032071-3

• Riley, Kenneth F.; Hobson, Michael P.; Bence, Stephen J. (1997), Mathematical methods for physics and engineering, Cambridge University Press, ISBN 0-521-55506-X

• Schiff, Leonard I. (1968), Quantum Mechanics (3rd ed.), McGraw–Hill

• Weinberg, Steven (1995), The Quantum Theory of Fields. Volume I: Foundations, Cambridge University Press, ISBN 0-521-55001-7

• Wherrett, Brian S. (1987), Group Theory for Atoms, Molecules and Solids, Prentice–Hall International, ISBN 0-13-365461-3

• Zabrodin, Anton; Brezin, Édouard; Kazakov, Vladimir; Serban, Didina; Wiegmann, Paul (2006), Applications of Random Matrices in Physics (NATO Science Series II: Mathematics, Physics and Chemistry), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-1-4020-4530-1 26.15. EXTERNAL LINKS 183

26.14.2 Historical references

• A. Cayley A memoir on the theory of matrices. Phil. Trans. 148 1858 17-37; Math. Papers II 475-496 • Bôcher, Maxime (2004), Introduction to higher algebra, New York, NY: Dover Publications, ISBN 978-0-486- 49570-5, reprint of the 1907 original edition • Cayley, Arthur (1889), The collected mathematical papers of Arthur Cayley, I (1841–1853), Cambridge Uni- versity Press, pp. 123–126 • Dieudonné, Jean, ed. (1978), Abrégé d'histoire des mathématiques 1700-1900, Paris, FR: Hermann • Hawkins, Thomas (1975), “Cauchy and the spectral theory of matrices”, Historia Mathematica 2: 1–29, doi:10.1016/0315-0860(75)90032-4, ISSN 0315-0860, MR 0469635 • Knobloch, Eberhard (1994), “From Gauss to Weierstrass: determinant theory and its historical evaluations”, The intersection of history and mathematics, Science Networks Historical Studies 15, Basel, Boston, Berlin: Birkhäuser, pp. 51–66, MR 1308079 • Kronecker, Leopold (1897), Hensel, Kurt, ed., Leopold Kronecker’s Werke, Teubner • Mehra, Jagdish; Rechenberg, Helmut (1987), The Historical Development of Quantum Theory (1st ed.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-96284-9 • Shen, Kangshen; Crossley, John N.; Lun, Anthony Wah-Cheung (1999), Nine Chapters of the Mathematical Art, Companion and Commentary (2nd ed.), Oxford University Press, ISBN 978-0-19-853936-0 • Weierstrass, Karl (1915), Collected works 3

26.15 External links

Encyclopedic articles

• Hazewinkel, Michiel, ed. (2001), “Matrix”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

History

• MacTutor: Matrices and determinants • Matrices and Linear Algebra on the Earliest Uses Pages • Earliest Uses of Symbols for Matrices and Vectors

Online books

• Kaw, Autar K., Introduction to Matrix Algebra, ISBN 978-0-615-25126-4 • The Matrix Cookbook (PDF), retrieved 24 March 2014 • Brookes, Mike (2005), The Matrix Reference Manual, London: Imperial College, retrieved 10 Dec 2008

Online matrix calculators

• SimplyMath (Matrix Calculator) • Matrix Calculator (DotNumerics) • Xiao, Gang, Matrix calculator, retrieved 10 Dec 2008 • Online matrix calculator, retrieved 10 Dec 2008 • Online matrix calculator (ZK framework), retrieved 26 Nov 2009 184 CHAPTER 26. MATRIX (MATHEMATICS)

• Oehlert, Gary W.; Bingham, Christopher, MacAnova, University of Minnesota, School of Statistics, retrieved 10 Dec 2008, a freeware package for matrix algebra and statistics • Online matrix calculator, retrieved 14 Dec 2009

• Operation with matrices in R (determinant, track, inverse, adjoint, transpose) Chapter 27

Origin (mathematics)

+y

-x +x

-y

The origin of a Cartesian coordinate system

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space.

185 186 CHAPTER 27. ORIGIN (MATHEMATICS)

27.1 Cartesian coordinates

In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.[1] The origin divides each of these axes into two halves, a positive and a negative semiaxis.[2] Points can then be located with reference to the origin by giving their numerical coordinates—that is, the positions of their projections along each axis, either in the positive or negative direction. The coordinates of the origin are always all zero, for example (0,0) in two dimensions and (0,0,0) in three.[1]

27.2 Other coordinate systems

In a polar coordinate system, the origin may also be called the pole. It does not itself have well-defined polar coor- dinates, because the polar coordinates of a point include the angle made by the positive x-axis and the ray from the origin to the point, and this ray is not well-defined for the origin itself.[3] In Euclidean geometry, the origin may be chosen freely as any convenient point of reference.[4] The origin of the complex plane can be referred as the point where real axis and imaginary axis intersect each other. In other words, it is the complex number zero.[5]

27.3 See also

• Null vector, an analogous point of a vector space • Point on plane closest to origin

• Radial basis function, a function depending only on the distance from the origin

27.4 References

[1] Madsen, David A. (2001), Engineering Drawing and Design, Delmar drafting series, Thompson Learning, p. 120, ISBN 9780766816343.

[2] Pontrjagin, Lev S. (1984), Learning higher mathematics, Springer series in Soviet mathematics, Springer-Verlag, p. 73, ISBN 9783540123514.

[3] Tanton, James Stuart (2005), Encyclopedia of Mathematics, Infobase Publishing, ISBN 9780816051243.

[4] Lee, John M. (2013), Axiomatic Geometry, Pure and Applied Undergraduate Texts 21, American Mathematical Society, p. 134, ISBN 9780821884782.

[5] Gonzalez, Mario (1991), Classical Complex Analysis, Chapman & Hall Pure and Applied Mathematics, CRC Press, ISBN 9780824784157. Chapter 28

Orthogonal matrix

In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e.

QTQ = QQT = I, where I is the identity matrix. This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse:

QT = Q−1, An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q∗) and therefore normal (Q∗Q = QQ∗) in the reals. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the dot product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation or reflection. In other words, it is a unitary transformation. The set of n × n orthogonal matrices forms a group O(n), known as the orthogonal group. The subgroup SO(n) consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation. The complex analogue of an orthogonal matrix is a unitary matrix.

28.1 Overview

An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. Although we consider only real matrices here, the definition can be used for matrices with entries from any field. However, orthogonal matrices arise naturally from dot products, and for matrices of complex numbers that leads instead to the unitary requirement. Orthogonal matrices preserve the dot product,[1] so, for vectors u, v in an n-dimensional real Euclidean space

u · v = (Qu) · (Qv) where Q is an orthogonal matrix. To see the inner product connection, consider a vector v in an n-dimensional real Euclidean space. Written with respect to an orthonormal basis, the squared length of v is vTv. If a linear transformation, in matrix form Qv, preserves vector lengths, then

vTv = (Qv)T(Qv) = vTQTQv. Thus finite-dimensional linear isometries—rotations, reflections, and their combinations—produce orthogonal ma- trices. The converse is also true: orthogonal matrices imply orthogonal transformations. However, linear algebra

187 188 CHAPTER 28. ORTHOGONAL MATRIX

includes orthogonal transformations between spaces which may be neither finite-dimensional nor of the same dimen- sion, and these have no orthogonal matrix equivalent. Orthogonal matrices are important for a number of reasons, both theoretical and practical. The n×n orthogonal matri- ces form a group under matrix multiplication, the orthogonal group denoted by O(n), which—with its subgroups—is widely used in mathematics and the physical sciences. For example, the point group of a molecule is a subgroup of O(3). Because floating point versions of orthogonal matrices have advantageous properties, they are key to many algo- rithms in numerical linear algebra, such as QR decomposition. As another example, with appropriate normalization the discrete cosine transform (used in MP3 compression) is represented by an orthogonal matrix.

28.2 Examples

Below are a few examples of small orthogonal matrices and possible interpretations. [ ] 1 0 • (transformation identity) 0 1

An instance of a 2×2 rotation matrix: [ ] [ ] cos θ − sin θ 0.96 −0.28 • R(16.26◦) = = ( by rotation16.26◦) sin θ cos θ 0.28 0.96 [ ] 1 0 • ( across reflectionx-axis) 0 −1   ( ) − − 0 0.80 0.60 rotoinversion: • 0.80 −0.36 0.48 − ◦ 0.60 0.48 −0.64 axis(0, 3/5, 4/5), angle 90   0 0 0 1 0 0 1 0 •   (axes coordinate of permutation) 1 0 0 0 0 1 0 0

28.3 Elementary constructions

28.3.1 Lower dimensions

The simplest orthogonal matrices are the 1×1 matrices [1] and [−1] which we can interpret as the identity and a reflection of the real line across the origin. The 2 × 2 matrices have the form

[ ] p t , q u which orthogonality demands satisfy the three equations

1 = p2 + t2, 1 = q2 + u2, 0 = pq + tu. In consideration of the first equation, without loss of generality let p = cos θ, q = sin θ; then either t = −q, u = p or t = q, u = −p. We can interpret the first case as a rotation by θ (where θ = 0 is the identity), and the second as a reflection across a line at an angle of θ/2. 28.3. ELEMENTARY CONSTRUCTIONS 189

[ ] [ ] cos θ − sin θ cos θ sin θ (rotation), (reflection) sin θ cos θ sin θ − cos θ

The special case of the reflection matrix with θ = 90° generates a reflection about the line at 45° given by y = x and therefore exchanges x and y; it is a permutation matrix, with a single 1 in each column and row (and otherwise 0):

[ ] 0 1 . 1 0

The identity is also a permutation matrix. A reflection is its own inverse, which implies that a reflection matrix is symmetric (equal to its transpose) as well as orthogonal. The product of two rotation matrices is a rotation matrix, and the product of two reflection matrices is also a rotation matrix.

28.3.2 Higher dimensions

Regardless of the dimension, it is always possible to classify orthogonal matrices as purely rotational or not, but for 3 × 3 matrices and larger the non-rotational matrices can be more complicated than reflections. For example,

    −1 0 0 0 −1 0  0 −1 0  and 1 0 0  0 0 −1 0 0 −1 represent an inversion through the origin and a rotoinversion about the z axis.

  cos(α) cos(γ) − sin(α) sin(β) sin(γ) − sin(α) cos(β) − cos(α) sin(γ) − sin(α) sin(β) cos(γ) cos(α) sin(β) sin(γ) + sin(α) cos(γ) cos(α) cos(β) cos(α) sin(β) cos(γ) − sin(α) sin(γ)  cos(β) sin(γ) − sin(β) cos(β) cos(γ)

Rotations become more complicated in higher dimensions; they can no longer be completely characterized by one angle, and may affect more than one planar subspace. It is common to describe a 3 × 3 rotation matrix in terms of an axis and angle, but this only works in three dimensions. Above three dimensions two or more angles are needed, each associated with a plane of rotation. However, we have elementary building blocks for permutations, reflections, and rotations that apply in general.

28.3.3 Primitives

The most elementary permutation is a transposition, obtained from the identity matrix by exchanging two rows. Any n×n permutation matrix can be constructed as a product of no more than n − 1 transpositions. A Householder reflection is constructed from a non-null vector v as

vvT Q = I − 2 . vTv Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of v. This is a reflection in the hyperplane perpendicular to v (negating any vector component parallel to v). If v is a unit vector, then Q = I − 2vvT suffices. A Householder reflection is typically used to simultaneously zero the lower part of a column. Any orthogonal matrix of size n × n can be constructed as a product of at most n such reflections. A Givens rotation acts on a two-dimensional (planar) subspace spanned by two coordinate axes, rotating by a chosen angle. It is typically used to zero a single subdiagonal entry. Any rotation matrix of size n×n can be constructed as a product of at most n(n − 1)/2 such rotations. In the case of 3 × 3 matrices, three such rotations suffice; and by fixing 190 CHAPTER 28. ORTHOGONAL MATRIX the sequence we can thus describe all 3 × 3 rotation matrices (though not uniquely) in terms of the three angles used, often called Euler angles. A Jacobi rotation has the same form as a Givens rotation, but is used to zero both off-diagonal entries of a 2 × 2 symmetric submatrix.

28.4 Properties

28.4.1 Matrix properties

A real square matrix is orthogonal if and only if its columns form an orthonormal basis of the Euclidean space Rn with the ordinary Euclidean dot product, which is the case if and only if its rows form an orthonormal basis of Rn. It might be tempting to suppose a matrix with orthogonal (not orthonormal) columns would be called an orthogonal matrix, but such matrices have no special interest and no special name; they only satisfy MTM = D, with D a diagonal matrix. The determinant of any orthogonal matrix is +1 or −1. This follows from basic facts about determinants, as follows:

1 = det(I) = det(QTQ) = det(QT) det(Q) = (det(Q))2. The converse is not true; having a determinant of ±1 is no guarantee of orthogonality, even with orthogonal columns, as shown by the following counterexample.

[ ] 2 0 1 0 2 With permutation matrices the determinant matches the signature, being +1 or −1 as the parity of the permutation is even or odd, for the determinant is an alternating function of the rows. Stronger than the determinant restriction is the fact that an orthogonal matrix can always be diagonalized over the complex numbers to exhibit a full set of eigenvalues, all of which must have (complex) modulus 1.

28.4.2 Group properties

The inverse of every orthogonal matrix is again orthogonal, as is the matrix product of two orthogonal matrices. In fact, the set of all n × n orthogonal matrices satisfies all the axioms of a group. It is a compact Lie group of dimension n(n − 1)/2, called the orthogonal group and denoted by O(n). The orthogonal matrices whose determinant is +1 form a path-connected normal subgroup of O(n) of index 2, the special orthogonal group SO(n) of rotations. The quotient group O(n)/SO(n) is isomorphic to O(1), with the projection map choosing [+1] or [−1] according to the determinant. Orthogonal matrices with determinant −1 do not include the identity, and so do not form a subgroup but only a coset; it is also (separately) connected. Thus each orthogonal group falls into two pieces; and because the projection map splits, O(n) is a semidirect product of SO(n) by O(1). In practical terms, a comparable statement is that any orthogonal matrix can be produced by taking a rotation matrix and possibly negating one of its columns, as we saw with 2×2 matrices. If n is odd, then the semidirect product is in fact a direct product, and any orthogonal matrix can be produced by taking a rotation matrix and possibly negating all of its columns. This follows from the property of determinants that negating a column negates the determinant, and thus negating an odd (but not even) number of columns negates the determinant. Now consider (n + 1) × (n + 1) orthogonal matrices with bottom right entry equal to 1. The remainder of the last column (and last row) must be zeros, and the product of any two such matrices has the same form. The rest of the matrix is an n × n orthogonal matrix; thus O(n) is a subgroup of O(n + 1) (and of all higher groups).

  0    .  O(n) .  0 0 ··· 0 1 28.4. PROPERTIES 191

Since an elementary reflection in the form of a Householder matrix can reduce any orthogonal matrix to this con- strained form, a series of such reflections can bring any orthogonal matrix to the identity; thus an orthogonal group is a reflection group. The last column can be fixed to any unit vector, and each choice gives a different copy of O(n) in O(n + 1); in this way O(n + 1) is a bundle over the unit sphere Sn with fiber O(n). Similarly, SO(n) is a subgroup of SO(n + 1); and any special orthogonal matrix can be generated by Givens plane rotations using an analogous procedure. The bundle structure persists: SO(n) ↪ SO(n + 1) → Sn. A single rotation can produce a zero in the first row of the last column, and series of n−1 rotations will zero all but the last row of the last column of an n × n rotation matrix. Since the planes are fixed, each rotation has only one degree of freedom, its angle. By induction, SO(n) therefore has

(n − 1) + (n − 2) + ··· + 1 = n(n − 1)/2

degrees of freedom, and so does O(n). Permutation matrices are simpler still; they form, not a Lie group, but only a finite group, the order n! symmetric group Sn. By the same kind of argument, Sn is a subgroup of Sn₊₁. The even permutations produce the subgroup of permutation matrices of determinant +1, the order n!/2 alternating group.

28.4.3 Canonical form

More broadly, the effect of any orthogonal matrix separates into independent actions on orthogonal two-dimensional subspaces. That is, if Q is special orthogonal then one can always find an orthogonal matrix P, a (rotational) change of basis, that brings Q into block diagonal form:

    R R 1 1  .  T  .  T  ..  P QP =  ..  (neven ),P QP =   (nodd ).  R  R k k 1

where the matrices R1, ..., Rk are 2 × 2 rotation matrices, and with the remaining entries zero. Exceptionally, a rotation block may be diagonal, ±I. Thus, negating one column if necessary, and noting that a 2 × 2 reflection diagonalizes to a +1 and −1, any orthogonal matrix can be brought to the form

  R1    ..   . 0    T  Rk  P QP =  ,  1   .   0 ..  1

The matrices R1, ..., Rk give conjugate pairs of eigenvalues lying on the unit circle in the complex plane; so this decomposition confirms that all eigenvalues have absolute value 1. If n is odd, there is at least one real eigenvalue, +1 or −1; for a 3 × 3 rotation, the eigenvector associated with +1 is the rotation axis.

28.4.4 Lie algebra

Suppose the entries of Q are differentiable functions of t, and that t = 0 gives Q = I. Differentiating the orthogonality condition

QTQ = I yields 192 CHAPTER 28. ORTHOGONAL MATRIX

Q˙ TQ + QTQ˙ = 0 Evaluation at t = 0 (Q = I) then implies

Q˙ T = −Q.˙ In Lie group terms, this means that the Lie algebra of an orthogonal matrix group consists of skew-symmetric matrices. Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal matrix (in fact, special orthogonal). For example, the three-dimensional object physics calls angular velocity is a differential rotation, thus a vector in the Lie algebra so(3) tangent to SO(3). Given ω = (xθ, yθ, zθ), with v = (x, y, z) being a unit vector, the correct skew-symmetric matrix form of ω is

  0 −zθ yθ Ω =  zθ 0 −xθ. −yθ xθ 0 The exponential of this is the orthogonal matrix for rotation around axis v by angle θ; setting c = cos θ/2, s = sin θ/2,

  1 − 2s2 + 2x2s2 2xys2 − 2zsc 2xzs2 + 2ysc exp(Ω) =  2xys2 + 2zsc 1 − 2s2 + 2y2s2 2yzs2 − 2xsc . 2xzs2 − 2ysc 2yzs2 + 2xsc 1 − 2s2 + 2z2s2

28.5 Numerical linear algebra

28.5.1 Benefits

Numerical analysis takes advantage of many of the properties of orthogonal matrices for numerical linear algebra, and they arise naturally. For example, it is often desirable to compute an orthonormal basis for a space, or an orthogonal change of bases; both take the form of orthogonal matrices. Having determinant ±1 and all eigenvalues of magnitude 1 is of great benefit for numeric stability. One implication is that the condition number is 1 (which is the minimum), so errors are not magnified when multiplying with an orthogonal matrix. Many algorithms use orthogonal matrices like Householder reflections and Givens rotations for this reason. It is also helpful that, not only is an orthogonal matrix invertible, but its inverse is available essentially free, by exchanging indices. Permutations are essential to the success of many algorithms, including the workhorse Gaussian elimination with partial pivoting (where permutations do the pivoting). However, they rarely appear explicitly as matrices; their special form allows more efficient representation, such as a list of n indices. Likewise, algorithms using Householder and Givens matrices typically use specialized methods of multiplication and storage. For example, a Givens rotation affects only two rows of a matrix it multiplies, changing a full multiplication of order n3 to a much more efficient order n. When uses of these reflections and rotations introduce zeros in a matrix, the space vacated is enough to store sufficient data to reproduce the transform, and to do so robustly. (Following Stewart (1976), we do not store a rotation angle, which is both expensive and badly behaved.)

28.5.2 Decompositions

A number of important matrix decompositions (Golub & Van Loan 1996) involve orthogonal matrices, including especially:

QR decomposition M = QR, Q orthogonal, R upper triangular Singular value decomposition M = UΣVT, U and V orthogonal, Σ non-negative diagonal Eigendecomposition of a symmetric matrix (decomposition according to the spectral theorem) S = QΛQT, S symmetric, Q orthogonal, Λ diagonal Polar decomposition M = QS, Q orthogonal, S symmetric non-negative definite 28.5. NUMERICAL LINEAR ALGEBRA 193

Examples

Consider an overdetermined system of linear equations, as might occur with repeated measurements of a physical phenomenon to compensate for experimental errors. Write Ax = b, where A is m × n, m > n. A QR decomposition reduces A to upper triangular R. For example, if A is 5 × 3 then R has the form

  ⋆ ⋆ ⋆   0 ⋆ ⋆   R = 0 0 ⋆. 0 0 0 0 0 0

The linear least squares problem is to find the x that minimizes ‖Ax − b‖, which is equivalent to projecting b to the subspace spanned by the columns of A. Assuming the columns of A (and hence R) are independent, the projection solution is found from ATAx = ATb. Now ATA is square (n × n) and invertible, and also equal to RTR. But the lower rows of zeros in R are superfluous in the product, which is thus already in lower-triangular upper-triangular factored form, as in Gaussian elimination (Cholesky decomposition). Here orthogonality is important not only for reducing ATA = (RTQT)QR to RTR, but also for allowing solution without magnifying numerical problems. In the case of a linear system which is underdetermined, or an otherwise non-invertible matrix, singular value de- composition (SVD) is equally useful. With A factored as UΣVT, a satisfactory solution uses the Moore-Penrose pseudoinverse, VΣ+UT, where Σ+ merely replaces each non-zero diagonal entry with its reciprocal. Set x to VΣ+UTb. The case of a square invertible matrix also holds interest. Suppose, for example, that A is a 3 × 3 rotation matrix which has been computed as the composition of numerous twists and turns. Floating point does not match the mathematical ideal of real numbers, so A has gradually lost its true orthogonality. A Gram-Schmidt process could orthogonalize the columns, but it is not the most reliable, nor the most efficient, nor the most invariant method. The polar decomposition factors a matrix into a pair, one of which is the unique closest orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular. (Closeness can be measured by any matrix norm invariant under an orthogonal change of basis, such as the spectral norm or the Frobenius norm.) For a near-orthogonal matrix, rapid convergence to the orthogonal factor can be achieved by a "Newton’s method" approach due to Higham (1986) (1990), repeatedly averaging the matrix with its inverse transpose. Dubrulle (1994) has published an accelerated method with a convenient convergence test. For example, consider a non-orthogonal matrix for which the simple averaging algorithm takes seven steps

[ ] [ ] [ ] − 3 1 → 1.8125 0.0625 → · · · → 0.8 0.6 7 5 3.4375 2.6875 0.6 0.8

and which acceleration trims to two steps (with γ = 0.353553, 0.565685).

[ ] [ ] [ ] − − 3 1 → 1.41421 1.06066 → 0.8 0.6 7 5 1.06066 1.41421 0.6 0.8

Gram-Schmidt yields an inferior solution, shown by a Frobenius distance of 8.28659 instead of the minimum 8.12404.

[ ] [ ] − 3 1 → 0.393919 0.919145 7 5 0.919145 0.393919

28.5.3 Randomization

Some numerical applications, such as Monte Carlo methods and exploration of high-dimensional data spaces, require generation of uniformly distributed random orthogonal matrices. In this context, “uniform” is defined in terms of Haar measure, which essentially requires that the distribution not change if multiplied by any freely chosen orthogonal matrix. Orthogonalizing matrices with independent uniformly distributed random entries does not result in uniformly distributed orthogonal matrices, but the QR decomposition of independent normally distributed random entries does, 194 CHAPTER 28. ORTHOGONAL MATRIX

as long as the diagonal of R contains only positive entries. Stewart (1980) replaced this with a more efficient idea that Diaconis & Shahshahani (1987) later generalized as the “subgroup algorithm” (in which form it works just as well for permutations and rotations). To generate an (n + 1) × (n + 1) orthogonal matrix, take an n × n one and a uniformly distributed unit vector of dimension n + 1. Construct a Householder reflection from the vector, then apply it to the smaller matrix (embedded in the larger size with a 1 at the bottom right corner).

28.5.4 Nearest orthogonal matrix

The problem of finding the orthogonal matrix Q nearest a given matrix M is related to the Orthogonal Procrustes problem. There are several different ways to get the unique solution, the simplest of which is taking the singular value decomposition of M and replacing the singular values with ones. Another method expresses the R explicitly but requires the use of a matrix square root:[2]

T − 1 Q = M(M M) 2

This may be combined with the Babylonian method for extracting the square root of a matrix to give a recurrence which converges to an orthogonal matrix quadratically:

−1 T −1 Qn+1 = 2M(Qn M + M Qn)

[3] where Q0 = M . These iterations are stable provided the condition number of M is less than three.

28.6 Spin and pin

A subtle technical problem afflicts some uses of orthogonal matrices. Not only are the group components with determinant +1 and −1 not connected to each other, even the +1 component, SO(n), is not simply connected (except for SO(1), which is trivial). Thus it is sometimes advantageous, or even necessary, to work with a covering group of SO(n), the spin group, Spin(n). Likewise, O(n) has covering groups, the pin groups, Pin(n). For n > 2, Spin(n) is simply connected and thus the universal covering group for SO(n). By far the most famous example of a spin group is Spin(3), which is nothing but SU(2), or the group of unit quaternions. The Pin and Spin groups are found within Clifford algebras, which themselves can be built from orthogonal matrices.

28.7 Rectangular matrices

If Q is not a square matrix, then the conditions QTQ = I and QQT = I are not equivalent. The condition QTQ = I says that the columns of Q are orthonormal. This can only happen if Q is an m × n matrix with n ≤ m. Similarly, QQT = I says that the rows of Q are orthonormal, which requires n ≥ m. There is no standard terminology for these matrices. They are sometimes called “orthonormal matrices”, sometimes “orthogonal matrices”, and sometimes simply “matrices with orthonormal rows/columns”.

28.8 See also

• Orthogonal group

• Rotation (mathematics)

• Skew-symmetric matrix, a matrix whose transpose is its negative

• Symplectic matrix

• Unitary matrix 28.9. NOTES 195

28.9 Notes

[1] “Paul’s online math notes”, Paul Dawkins, Lamar University, 2008. Theorem 3(c)

[2] “Finding the Nearest Orthonormal Matrix”, Berthold K. P. Horn, MIT.

[3] “Newton’s Method for the Matrix Square Root”, Nicholas J. Higham, Mathematics of Computation, Volume 46, Number 174, 1986.

28.10 References

• Diaconis, Persi; Shahshahani, Mehrdad (1987), “The subgroup algorithm for generating uniform random vari- ables”, Prob. In Eng. And Info. Sci. 1: 15–32, doi:10.1017/S0269964800000255, ISSN 0269-9648 • Dubrulle, Augustin A. (1999), “An Optimum Iteration for the Matrix Polar Decomposition”, Elect. Trans. Num. Anal. 8: 21–25 • Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3/e ed.), Baltimore: Johns Hopkins University Press, ISBN 978-0-8018-5414-9 • Higham, Nicholas (1986), “Computing the Polar Decomposition—with Applications”, SIAM Journal on Sci- entific and Statistical Computing 7 (4): 1160–1174, doi:10.1137/0907079, ISSN 0196-5204 • Higham, Nicholas; Schreiber, Robert (July 1990), “Fast polar decomposition of an arbitrary matrix”, SIAM Journal on Scientific and Statistical Computing 11 (4): 648–655, doi:10.1137/0911038, ISSN 0196-5204 • Stewart, G. W. (1976), “The Economical Storage of Plane Rotations”, Numerische Mathematik 25 (2): 137– 138, doi:10.1007/BF01462266, ISSN 0029-599X • Stewart, G. W. (1980), “The Efficient Generation of Random Orthogonal Matrices with an Application to Condition Estimators”, SIAM J. Numer. Anal. 17 (3): 403–409, doi:10.1137/0717034, ISSN 0036-1429

28.11 External links

• Hazewinkel, Michiel, ed. (2001), “Orthogonal matrix”, Encyclopedia of Mathematics, Springer, ISBN 978-1- 55608-010-4

• Tutorial and Interactive Program on Orthogonal Matrix Chapter 29

Outline of calculus

The following outline is provided as an overview of and topical guide to calculus: Calculus – branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. Calculus is the study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.

29.1 Branches of calculus

• Differential calculus

• Integral calculus

29.2 History of calculus

History of calculus

29.3 General calculus concepts

• Fundamental theorem of calculus

• Differential calculus

• Integral calculus

• Law of continuity

• Limits of integration

• List of calculus topics

• Important publications in calculus

• Mathematics

• Multivariable calculus

• Non-standard analysis

• Partial derivative

196 29.4. CALCULUS SCHOLARS 197

29.4 Calculus scholars

Sir Isaac Newton - Gottfried Leibniz

29.5 Calculus lists

List of calculus topics

• Table of mathematical symbols

29.6 See also

• Table of mathematical symbols

29.7 References

[1] Latorre, Donald R.; Kenelly, John W.; Reed, Iris B.; Biggers, Sherry (2007), Calculus Concepts: An Applied Approach to the Mathematics of Change, Cengage Learning, p. 2, ISBN 0-618-78981-2, Chapter 1, p 2

29.8 External links

• Weisstein, Eric W., “Calculus”, MathWorld. • Topics on Calculus at PlanetMath.org.

• Calculus Made Easy (1914) by Silvanus P. Thompson Full text in PDF

• Calculus.org: The Calculus page at University of California, Davis – contains resources and links to other sites • COW: Calculus on the Web at Temple University - contains resources ranging from pre-calculus and associated algebra • Online Integrator (WebMathematica) from Wolfram Research

• The Role of Calculus in College Mathematics from ERICDigests.org • OpenCourseWare Calculus from the Massachusetts Institute of Technology

• Infinitesimal Calculus – an article on its historical development, in Encyclopaedia of Mathematics, Michiel Hazewinkel ed. . Chapter 30

Parallelogram

This article is about the quadrilateral shape. For the album by Linda Perhacs, see Parallelograms (album).

In Euclidean geometry, a parallelogram is a (non self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. The three-dimensional counterpart of a parallelogram is a parallelepiped. A quadrilateral with one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The etymology (in Greek παραλληλ-όγραμμον, a shape “of parallel lines”) reflects the definition.

30.1 Special cases

• Rhomboid – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles[1] • Rectangle – A parallelogram with four angles of equal size • Rhombus – A parallelogram with four sides of equal length. • Square – A parallelogram with four sides of equal length and angles of equal size (right angles).

30.2 Characterizations

A simple (non self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true:[2][3]

• Two pairs of opposite sides are equal in length. • Two pairs of opposite angles are equal in measure. • The diagonals bisect each other. • One pair of opposite sides are parallel and equal in length. • Adjacent angles are supplementary. • Each diagonal divides the quadrilateral into two congruent triangles. • The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law.) • It has rotational symmetry of order 2.

198 30.3. PROPERTIES 199

Quadrilaterals by symmetry

• The sum of the distances from any interior point to the sides is independent of the location of the point.[4] (This is an extension of Viviani’s theorem.)

Note that this means that all parallelograms have all the properties listed above, and conversely, if just one of these statements is true in a convex quadrilateral, then it is a parallelogram.

30.3 Properties

• Diagonals of a parallelogram bisect each other,

• Opposite sides of a parallelogram are parallel (by definition) and so will never intersect.

• The area of a parallelogram is twice the area of a triangle created by one of its diagonals. 200 CHAPTER 30. PARALLELOGRAM

• The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides. • Any line through the midpoint of a parallelogram bisects the area.[5] • Any non-degenerate affine transformation takes a parallelogram to another parallelogram. • A parallelogram has rotational symmetry of order 2 (through 180°). If it also has two lines of reflectional symmetry then it must be a rhombus or an oblong. • The perimeter of a parallelogram is 2(a + b) where a and b are the lengths of adjacent sides. • Unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area.[6]

30.4 Area formula

• A parallelogram with base b and height h can be divided into a trapezoid and a right triangle, and rearranged into a rectangle, as shown in the figure to the left. This means that the area of a parallelogram is the same as that of a rectangle with the same base and height:

K = bh.

• The base × height area formula can also be derived using the figure to the right. The area K of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles.

The area of the rectangle is

Krect = (B + A) × H

and the area of a single orange triangle is 1 K = A × H. tri 2

Therefore, the area of the parallelogram is

K = Krect − 2 × Ktri = ((B + A) × H) − (A × H) = B × H.

• Another area formula, for two sides B and C and angle θ, is

K = B · C · sin θ.

• The area of a parallelogram with sides B and C (B ≠ C) and angle γ at the intersection of the diagonals is given by[7]

| tan γ| K = · B2 − C2 . 2

• When the parallelogram is specified from the lengths B and C of two adjacent sides together with the length D1 of either diagonal, then the area can be found from Heron’s formula. Specifically it is

√ K = 2 S(S − B)(S − C)(S − D1)

where S = (B + C + D1)/2 and the leading factor 2 comes from the fact that the number of congruent triangles that the chosen diagonal divides the parallelogram into is two. 30.4. AREA FORMULA 201

A parallelogram can be rearranged into a rectangle with the same area.

30.4.1 Another parallelogram with the same base and height

Given any parallelogram divided in half through one of the diagonals, one can move one of the triangles to the other side of the other triangle to get another parallelogram with the same base and height, and thus the same area. 202 CHAPTER 30. PARALLELOGRAM

H

θ

B A

The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram

30.4.2 Area in terms of Cartesian coordinates of vertices [ ] a a Let vectors a, b ∈ R2 and let V = 1 2 ∈ R2×2 denote the matrix with elements of a and b. Then the area of b1 b2 the parallelogram generated by a and b is equal to | det(V )| = |a b − a b | . [ ] 1 2 2 1 a a . . . a Let vectors a, b ∈ Rn and let V = 1 2 n ∈ R2×n . Then the area of the parallelogram generated by √ b1 b2 . . . bn a and b is equal to det(VV T) . Let points a, b, c ∈ R2 . Then the area of the parallelogram with vertices at a, b and c is equivalent to the absolute value of the determinant of a matrix built using a, b and c as rows with the last column padded using ones as follows:

 

a1 a2 1   K = det b1 b2 1 .

c1 c2 1

30.5 Proof that diagonals bisect each other

To prove that the diagonals of a parallelogram bisect each other, we will use congruent triangles: ∼ ∠ABE = ∠CDE (alternate interior angles are equal in measure) ∼ ∠BAE = ∠DCE (alternate interior angles are equal in measure).

(since these are angles that a transversal makes with parallel lines AB and DC). Also, side AB is equal in length to side DC, since opposite sides of a parallelogram are equal in length. Therefore triangles ABE and CDE are congruent (ASA postulate, two corresponding angles and the included side). Therefore,

AE = CE 30.6. PARALLELOGRAMS ARISING FROM OTHER FIGURES 203

A B

E

D C

Parallelogram ABCD

BE = DE. Since the diagonals AC and BD divide each other into segments of equal length, the diagonals bisect each other. Separately, since the diagonals AC and BD bisect each other at point E, point E is the midpoint of each diagonal.

30.6 Parallelograms arising from other figures

30.6.1 Automedian triangle

An automedian triangle is one whose medians are in the same proportions as its sides (though in a different order). If ABC is an automedian triangle in which vertex A stands opposite the side a, G is the centroid (where the three medians of ABC intersect), and AL is one of the extended medians of ABC with L lying on the circumcircle of ABC, then BGCL is a parallelogram.

30.6.2 Varignon parallelogram

The midpoints of the sides of an arbitrary quadrilateral are the vertices of a parallelogram, called its Varignon par- allelogram. If the quadrilateral is convex or concave (that is, not self-intersecting), then the area of the Varignon parallelogram is half the area of the quadrilateral.

30.6.3 Tangent parallelogram of an ellipse

For an ellipse, two diameters are said to be conjugate if and only if the tangent line to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding 204 CHAPTER 30. PARALLELOGRAM tangent parallelogram, sometimes called a bounding parallelogram, formed by the tangent lines to the ellipse at the four endpoints of the conjugate diameters. All tangent parallelograms for a given ellipse have the same area. It is possible to reconstruct an ellipse from any pair of conjugate diameters, or from any tangent parallelogram.

30.6.4 Faces of a parallelepiped

A parallelepiped is a three-dimensional figure whose six faces are parallelograms.

30.7 See also

• Fundamental parallelogram

30.8 References

[1] http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf

[2] Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry, Mathematical Association of America, 2010, pp. 51-52.

[3] Zalman Usiskin and Jennifer Griffin, “The Classification of Quadrilaterals. A Study of Definition”, Information Age Publishing, 2008, p. 22.

[4] Chen, Zhibo, and Liang, Tian. “The converse of Viviani’s theorem”, The College Mathematics Journal 37(5), 2006, pp. 390–391.

[5] Dunn, J.A., and J.E. Pretty, “Halving a triangle”, Mathematical Gazette 56, May 1972, p. 105.

[6] Weisstein, Eric W. “Triangle Circumscribing”. Wolfram Math World.

[7] Mitchell, Douglas W., “The area of a quadrilateral”, Mathematical Gazette, July 2009.

30.9 External links

• Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)

• Weisstein, Eric W., “Parallelogram”, MathWorld. • Interactive Parallelogram --sides, angles and slope

• Area of Parallelogram at cut-the-knot • Equilateral Triangles On Sides of a Parallelogram at cut-the-knot

• Definition and properties of a parallelogram with animated applet • Interactive applet showing parallelogram area calculation interactive applet Chapter 31

Position (vector)

In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point P in space in relation to an arbitrary reference origin O. Usually denoted x, r, or s, it corresponds to the straight-line distance from O to P:[1]

−−→ r = OP.

The term “position vector” is used mostly in the fields of differential geometry, mechanics and occasionally in vector calculus. Frequently this is used in two-dimensional or three-dimensional space, but can be easily generalized to Euclidean spaces in any number of dimensions.[2]

31.1 Definition

31.1.1 Three dimensions

In three dimensions, any set of three dimensional coordinates and their corresponding basis vectors can be used to define the location of a point in space—whichever is the simplest for the task at hand may be used. Commonly, one uses the familiar Cartesian coordinate system, or sometimes spherical polar coordinates, or cylindrical coordinates;

r(t) ≡ r (x, y, z) ≡ x(t)^ex + y(t)^ey + z(t)^ez

≡ r (r, θ, ϕ) ≡ r(t)^er(θ(t), ϕ(t))

≡ r (r, θ, z) ≡ r(t)^er(θ(t)) + z(t)^ez ··· where t is a parameter, owing to their rectangular or circular symmetry. These different coordinates and corresponding basis vectors represent the same position vector. More general curvilinear coordinates could be used instead, and are in contexts like continuum mechanics and general relativity (in the latter case one needs an additional time coordinate).

31.1.2 n dimensions

Linear algebra allows for the abstraction of an n-dimensional position vector. A position vector can be expressed as a linear combination of basis vectors:[3][4]

∑n r = xiei = x1e1 + x2e2 + ··· + xnen i=1

205 206 CHAPTER 31. POSITION (VECTOR)

t

r(t) p t a p − a

t = c

Space curve in 3D. The position vector r is parameterized by a scalar t. At r = a the red line is the tangent to the curve, and the blue plane is normal to the curve.

The set of all position vectors form position space (a vector space whose elements are the position vectors), since positions can be added (vector addition) and scaled in length (scalar multiplication) to obtain another position vector in the space. The notion of “space” is intuitive since each xi (i = 1, 2, …, n) can be any value, the collection of values defines a point in space. The dimension of the position space is n (also denoted dim(R) = n). The coordinates of the vector r with respect to the basis vectors ei are xi. The vector of coordinates forms the coordinate vector or n-tuple (x1, x2, …, xn). Each coordinate xi may be parameterized a number of parameters t. One parameter xi(t) would describe a curved 1D path, two parameters xi(t1, t2) describes a curved 2D surface, three xi(t1, t2, t3) describes a curved 3D volume of space, and so on. 31.2. APPLICATIONS 207

The linear span of a basis set B = {e1, e2, …, en} equals the position space R, denoted span(B) = R.

31.2 Applications

31.2.1 Differential geometry

Main article: Differential geometry

Position vector fields are used to describe continuous and differentiable space curves, in which case the independent parameter needs not be time, but can be (e.g.) arc length of the curve.

31.2.2 Mechanics

Main articles: Newtonian mechanics, Analytical mechanics and Equation of motion

In any equation of motion, the position vector r(t) is usually the most sought-after quantity because this function defines the motion of a particle (i.e. a point mass) - its location relative to a given coordinate system at some time t. To define motion in terms of position, each coordinate may be parametrized by time; since each successive value of time corresponds to a sequence of successive spatial locations given by the coordinates, so the continuum limit of many successive locations is a path the particle traces. In the case of one dimension, the position has only one component, so it effectively degenerates to a scalar coordinate. It could be, say, a vector in the x-direction, or the radial r-direction. Equivalent notations include: x ≡ x ≡ x(t), r ≡ r(t), s ≡ s(t) ···

31.3 Derivatives of position

a = dv dt v + dv dv

v m dr m v = dr dt r r + dr

Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a 208 CHAPTER 31. POSITION (VECTOR)

For a position vector r that is a function of time t, the time derivatives can be computed with respect to t. These derivatives have common utility in the study of kinematics, control theory, engineering and other sciences.

dr Velocity v = dt

where dr is an infinitesimally small displacement (vector).

dv d2r Acceleration a = dt = dt2

da d2v d3r Jerk j = dt = dt2 = dt3

These names for the first, second and third derivative of position are commonly used in basic kinematics.[5] By extension, the higher order derivatives can be computed in a similar fashion. Study of these higher order derivatives can improve approximations of the original displacement function. Such higher-order terms are required in order to accurately represent the displacement function as a sum of an infinite sequence, enabling several analytical techniques in engineering and physics.

31.4 Relationship to displacement vectors

A displacement vector can be defined as the action of uniformly translating spatial points in a given direction over a given distance. Thus the addition of displacement vectors expresses the composition of these displacement actions and scalar multiplication as scaling of the distance. With this in mind we may then define a position vector of a point in space as the displacement vector mapping a given origin to that point. Note thus position vectors depend on a choice of origin for the space, as well as displacement vectors depend on the choice of an initial point.

31.5 See also

• Affine space • Curve • Displacement vector • Line element • Parametric surface • Point (geometry)

31.6 Notes

[1] H.D. Young, R.A. Freedman (2008). University Physics (12th ed.). Addison-Wesley (Pearson International). ISBN 0-321- 50130-6.

[2] Keller, F. J, Gettys, W. E. et al. (1993), p 28–29

[3] Riley, K.F.; Hobson, M.P.; Bence, S.J. (2010). Mathematical methods for physics and engineering. Cambridge University Press. ISBN 978-0-521-86153-3.

[4] Lipschutz, S.; Lipson, M. (2009). Linear Algebra. McGraw Hill. ISBN 978-0-07-154352-1.

[5] Stewart, James (2001). "§2.8 - The Derivative As A Function”. Calculus (2nd ed.). Brooks/Cole. ISBN 0-534-37718-1.

31.7 References

1. Keller, F. J, Gettys, W. E. et al. (1993). “Physics: Classical and modern” 2nd ed. McGraw Hill Publishing Chapter 32

Reflection (mathematics)

This article is about reflection in geometry. For reflexivity of binary relations, see reflexive relation. In mathematics, a reflection (also spelled reflexion)[1] is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. The term “reflection” is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. Such isometries have a set of fixed points (the “mirror”) that is an affine subspace, but is possibly smaller than a hyperplane. For instance a reflection through a point is an involutive isometry with just one fixed point; the image of the letter p under it would look like a d. This operation is also known as a central inversion (Coxeter 1969, §7.2), and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term “reflection” means reflection in a hyperplane. A figure that does not change upon undergoing a reflection is said to have reflectional symmetry. Some mathematicians use "flip" as a synonym for “reflection”.[2][3][4]

32.1 Construction

In plane (or 3-dimensional) geometry, to find the reflection of a point one drops a perpendicular from the point onto the line (plane) used for reflection, and continues it to the same distance on the other side. To find the reflection of a figure, one reflects each point in the figure. To reflect point P in the line AB using compass and straightedge, proceed as follows (see figure):

• Step 1 (red): construct a circle with center at P and some fixed radius r to create points A′ and B′ on the line AB, which will be equidistant from P. • Step 2 (green): construct circles centered at A′ and B′ having radius r. P and Q will be the points of intersection of these two circles.

Point Q is then the reflection of point P in line AB.

32.2 Properties

The matrix for a reflection is orthogonal with determinant −1 and eigenvalues −1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents a rotation. Every rotation is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every improper rotation is the result of reflecting

209 210 CHAPTER 32. REFLECTION (MATHEMATICS)

B’C’

A’

M/2

M

CB

A

A reflection through an axis followed by a reflection across a second axis parallel to the first one results in a total motion that is a translation.

in an odd number. Thus reflections generate the orthogonal group, and this result is known as the Cartan–Dieudonné theorem. Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes. In general, a group generated by reflections in affine hyperplanes is known as a reflection group. The finite groups generated in this way are examples of Coxeter groups.

32.3 Reflection across a line in the plane

For more details on reflection of light rays, see Specular reflection § Direction of reflection.

Reflection across a line through the origin in two dimensions can be described by the following formula

v · l Ref (v) = 2 l − v l l · l 32.4. REFLECTION THROUGH A HYPERPLANE IN N DIMENSIONS 211

P

A'O B' A B

Q

Point Q is reflection of point P in the line AB.

Where v denotes the vector being reflected, l denotes any vector in the line being reflected in, and v·l denotes the dot product of v with l. Note the formula above can also be described as

− Refl(v) = 2Projl(v) v

Where the reflection of line l on a is equal to 2 times the projection of v on line l minus v. Reflections in a line have the eigenvalues of 1, and −1.

32.4 Reflection through a hyperplane in n dimensions

Given a vector a in Euclidean space Rn, the formula for the reflection in the hyperplane through the origin, orthogonal to a, is given by

v · a Ref (v) = v − 2 a a a · a where v ⋅ a denotes the dot product of v with a. Note that the second term in the above equation is just twice the vector projection of v onto a. One can easily check that

• Refa(v) = −v, if v is parallel to a, and

• Refa(v) = v, if v is perpendicular to a. 212 CHAPTER 32. REFLECTION (MATHEMATICS)

C

C' A B' B

A'

θ/2 θ

A reflection across an axis followed by a reflection in a second axis not parallel to the first one results in a total motion that is a rotation around the point of intersection of the axes.

Using the geometric product, the formula is

ava Ref (v) = − . a a2 Since these reflections are isometries of Euclidean space fixing the origin they may be represented by orthogonal matrices. The orthogonal matrix corresponding to the above reflection is the matrix whose entries are

aiaj Rij = δij − 2 ∥a∥2

where δij is the Kronecker delta. The formula for the reflection in the affine hyperplane v · a = c not through the origin is

v · a − c Ref (v) = v − 2 a. a,c a · a 32.5. SEE ALSO 213

32.5 See also

• Coordinate rotations and reflections

• Householder transformation • Inversive geometry

• Point reflection

• Plane of rotation • Reflection mapping

• Reflection group • Specular reflection

32.6 Notes

[1] “Reflexion” is an archaic spelling.

[2] Childs, Lindsay N. (2009), A Concrete Introduction to Higher Algebra (3rd ed.), Springer Science & Business Media, p. 251

[3] Gallian, Joseph (2012), Contemporary Abstract Algebra (8th ed.), Cengage Learning, p. 32

[4] Isaacs, I. Martin (1994), Algebra: A Graduate Course, American Mathematical Society, p. 6

32.7 References

• Coxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50458-0, MR 123930 • Popov, V.L. (2001), “Reflection”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978- 1-55608-010-4

• Weisstein, Eric W., “Reflection”, MathWorld.

32.8 External links

• Reflection in Line at cut-the-knot

• Understanding 2D Reflection and Understanding 3D Reflection by Roger Germundsson, The Wolfram Demon- strations Project. Chapter 33

Rigid body

The position of a rigid body is determined by the position of its center of mass and by its attitude (at least six parameters in total).[1]

In physics, a rigid body is an idealization of a solid body in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it. Even though such an object cannot physically exist due to relativity, objects can normally be assumed to be perfectly rigid if they are not moving near the speed of light. In classical mechanics a rigid body is usually considered as a continuous mass distribution, while in quantum mechan- ics a rigid body is usually thought of as a collection of point masses. For instance, in quantum mechanics molecules (consisting of the point masses: electrons and nuclei) are often seen as rigid bodies (see classification of molecules as rigid rotors).

214 33.1. KINEMATICS 215

33.1 Kinematics

33.1.1 Linear and angular position

The position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, namely that all its particles maintain the same distance relative to each other. If the body is rigid, it is sufficient to describe the position of at least three non-collinear particles. This makes it possible to reconstruct the position of all the other particles, provided that their time-invariant position relative to the three selected particles is known. However, typically a different, mathematically more convenient, but equivalent approach is used. The position of the whole body is represented by:

1. the linear position or position of the body, namely the position of one of the particles of the body, specifically chosen as a reference point (typically coinciding with the center of mass or centroid of the body), together with

2. the angular position (also known as orientation, or attitude) of the body.

Thus, the position of a rigid body has two components: linear and angular, respectively.[2] The same is true for other kinematic and kinetic quantities describing the motion of a rigid body, such as linear and angular velocity, acceleration, momentum, impulse, and kinetic energy.[3] The linear position can be represented by a vector with its tail at an arbitrary reference point in space (the origin of a chosen coordinate system) and its tip at an arbitrary point of interest on the rigid body, typically coinciding with its center of mass or centroid. This reference point may define the origin of a coordinate system fixed to the body. There are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles, a quaternion, or a direction cosine matrix (also referred to as a rotation matrix). All these methods actually define the orientation of a basis set (or coordinate system) which has a fixed orientation relative to the body (i.e. rotates together with the body), relative to another basis set (or coordinate system), from which the motion of the rigid body is observed. For instance, a basis set with fixed orientation relative to an airplane can be defined as a set of three orthogonal unit vectors b1, b2, b3, such that b1 is parallel to the chord line of the wing and directed forward, b2 is normal to the plane of symmetry and directed rightward, and b3 is given by the cross product b3 = b1 × b2 . In general, when a rigid body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translation and rotation, respectively. Indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation (roto-translation) of the body starting from a hypothetic reference position (not necessarily coinciding with a position actually taken by the body during its motion).

33.1.2 Linear and angular velocity

Velocity (also called linear velocity) and angular velocity are measured with respect to a frame of reference. The linear velocity of a rigid body is a vector quantity, equal to the time rate of change of its linear position. Thus, it is the velocity of a reference point fixed to the body. During purely translational motion (motion with no rotation), all points on a rigid body move with the same velocity. However, when motion involves rotation, the instantaneous velocity of any two points on the body will generally not be the same. Two points of a rotating body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of rotation. Angular velocity is a vector quantity that describes the angular speed at which the orientation of the rigid body is changing and the instantaneous axis about which it is rotating (the existence of this instantaneous axis is guaranteed by the Euler’s rotation theorem). All points on a rigid body experience the same angular velocity at all times. During purely rotational motion, all points on the body change position except for those lying on the instantaneous axis of rotation. The relationship between orientation and angular velocity is not directly analogous to the relationship between position and velocity. Angular velocity is not the time rate of change of orientation, because there is no such concept as an orientation vector that can be differentiated to obtain the angular velocity.

33.2 Kinematical equations 216 CHAPTER 33. RIGID BODY

33.2.1 Addition theorem for angular velocity

The angular velocity of a rigid body B in a reference frame N is equal to the sum of the angular velocity of a rigid body D in N and the angular velocity of B with respect to D:[4]

NωB = NωD + DωB.

In this case, rigid bodies and reference frames are indistinguishable and completely interchangeable.

33.2.2 Addition theorem for position

For any set of three points P, Q, and R, the position vector from P to R is the sum of the position vector from P to Q and the position vector from Q to R: rPR = rPQ + rQR.

33.2.3 Mathematical definition of velocity

The velocity of point P in reference frame N is defined using the time derivative in N of the position vector from O to P:[5]

Nd NvP = (rOP) dt where O is any arbitrary point fixed in reference frame N, and the N to the left of the d/dt operator indicates that the derivative is taken in reference frame N. The result is independent of the selection of O so long as O is fixed in N.

33.2.4 Mathematical definition of acceleration

The acceleration of point P in reference frame N is defined using the time derivative in N of its velocity:[5]

Nd NaP = (NvP). dt

33.2.5 Velocity of two points fixed on a rigid body

For two points P and Q that are fixed on a rigid body B, where B has an angular velocity NωB in the reference frame N, the velocity of Q in N can be expressed as a function of the velocity of P in N:[6]

NvQ = NvP + NωB × rPQ.

33.2.6 Acceleration of two points fixed on a rigid body

By differentiating the equation for the Velocity of two points fixed on a rigid body in N with respect to time, the acceleration in reference frame N of a point Q fixed on a rigid body B can be expressed as

( ) NaQ = NaP + NωB × NωB × rPQ + NαB × rPQ

[6] where NαB is the angular acceleration of B in the reference frame N. 33.2. KINEMATICAL EQUATIONS 217

33.2.7 Angular velocity and acceleration of two points fixed on a rigid body

As mentioned above, all points on a rigid body B have the same angular velocity NωB in a fixed reference frame N, and thus the same angular acceleration NαB.

33.2.8 Velocity of one point moving on a rigid body

If the point R is moving in rigid body B while B moves in reference frame N, then the velocity of R in N is

NvR = NvQ + BvR where Q is the point fixed in B that is instantaneously coincident with R at the instant of interest.[7] This relation is often combined with the relation for the Velocity of two points fixed on a rigid body.

33.2.9 Acceleration of one point moving on a rigid body

The acceleration in reference frame N of the point R moving in body B while B is moving in frame N is given by

NaR = NaQ + BaR + 2NωB × BvR where Q is the point fixed in B that instantaneously coincident with R at the instant of interest.[7] This equation is often combined with Acceleration of two points fixed on a rigid body.

33.2.10 Other quantities

If C is the origin of a local coordinate system L, attached to the body,

• the spatial or twist acceleration of a rigid body is defined as the spatial acceleration of C (as opposed to material acceleration above);

− × × ψ(t, r0) = a(t, r0) ω(t) v(t, r0) = ψc(t) + α(t) A(t)r0 where

• r0 represents the position of the point/particle with respect to the reference point of the body in terms of the local coordinate system L (the rigidity of the body means that this does not depend on time) • A(t) is the orientation matrix, an orthogonal matrix with determinant 1, representing the orientation (angular position) of the local coordinate system L, with respect to the arbitrary reference orientation of another coor- dinate system G. Think of this matrix as three orthogonal unit vectors, one in each column, which define the orientation of the axes of L with respect to G. • ω(t) represents the angular velocity of the rigid body

• v(t, r0) represents the total velocity of the point/particle

• a(t, r0) represents the total acceleration of the point/particle • α(t) represents the angular acceleration of the rigid body

• ψ(t, r0) represents the spatial acceleration of the point/particle • ψc(t) represents the spatial acceleration of the rigid body (i.e. the spatial acceleration of the origin of L).

In 2D, the angular velocity is a scalar, and matrix A(t) simply represents a rotation in the xy-plane by an angle which is the integral of the angular velocity over time. Vehicles, walking people, etc., usually rotate according to changes in the direction of the velocity: they move forward with respect to their own orientation. Then, if the body follows a closed orbit in a plane, the angular velocity integrated over a time interval in which the orbit is completed once, is an integer times 360°. This integer is the winding number with respect to the origin of the velocity. Compare the amount of rotation associated with the vertices of a polygon. 218 CHAPTER 33. RIGID BODY

33.3 Kinetics

Main article: Rigid body dynamics

Any point that is rigidly connected to the body can be used as reference point (origin of coordinate system L) to describe the linear motion of the body (the linear position, velocity and acceleration vectors depend on the choice). However, depending on the application, a convenient choice may be:

• the center of mass of the whole system, which generally has the simplest motion for a body moving freely in space; • a point such that the translational motion is zero or simplified, e.g. on an axle or hinge, at the center of a ball and socket joint, etc.

When the center of mass is used as reference point:

• The (linear) momentum is independent of the rotational motion. At any time it is equal to the total mass of the rigid body times the translational velocity. • The angular momentum with respect to the center of mass is the same as without translation: at any time it is equal to the inertia tensor times the angular velocity. When the angular velocity is expressed with respect to a coordinate system coinciding with the principal axes of the body, each component of the angular momentum is a product of a moment of inertia (a principal value of the inertia tensor) times the corresponding component of the angular velocity; the torque is the inertia tensor times the angular acceleration. • Possible motions in the absence of external forces are translation with constant velocity, steady rotation about a fixed principal axis, and also torque-free precession. • The net external force on the rigid body is always equal to the total mass times the translational acceleration (i.e., Newton’s second law holds for the translational motion, even when the net external torque is nonzero, and/or the body rotates). • The total kinetic energy is simply the sum of translational and rotational energy.

33.4 Geometry

Two rigid bodies are said to be different (not copies) if there is no proper rotation from one to the other. A rigid body is called chiral if its mirror image is different in that sense, i.e., if it has either no symmetry or its symmetry group contains only proper rotations. In the opposite case an object is called achiral: the mirror image is a copy, not a different object. Such an object may have a symmetry plane, but not necessarily: there may also be a plane of reflection with respect to which the image of the object is a rotated version. The latter applies for S2n, of which the case n = 1 is inversion symmetry. For a (rigid) rectangular transparent sheet, inversion symmetry corresponds to having on one side an image without rotational symmetry and on the other side an image such that what shines through is the image at the top side, upside down. We can distinguish two cases:

• the sheet surface with the image is not symmetric - in this case the two sides are different, but the mirror image of the object is the same, after a rotation by 180° about the axis perpendicular to the mirror plane. • the sheet surface with the image has a symmetry axis - in this case the two sides are the same, and the mirror image of the object is also the same, again after a rotation by 180° about the axis perpendicular to the mirror plane.

A sheet with a through and through image is achiral. We can distinguish again two cases:

• the sheet surface with the image has no symmetry axis - the two sides are different • the sheet surface with the image has a symmetry axis - the two sides are the same 33.5. CONFIGURATION SPACE 219

33.5 Configuration space

The configuration space of a rigid body with one point fixed (i.e., a body with zero translational motion) is given by the underlying manifold of the rotation group SO(3). The configuration space of a nonfixed (with non-zero trans- lational motion) rigid body is E+(3), the subgroup of direct isometries of the Euclidean group in three dimensions (combinations of translations and rotations).

33.6 See also

• Angular velocity

• Axes conventions

• Rigid body dynamics

• infinitesimal rotations

• Euler’s equations (rigid body dynamics)

• Euler’s laws

• Born rigidity

• Rigid rotor

• Geometric Mechanics

33.7 Notes

[1] Lorenzo Sciavicco, Bruno Siciliano (2000). "§2.4.2 Roll-pitch-yaw angles”. Modelling and control of robot manipulators (2nd ed.). Springer. p. 32. ISBN 1-85233-221-2.

[2] In general, the position of a point or particle is also known, in physics, as linear position, as opposed to the angular position of a line, or line segment (e.g., in circular motion, the “radius” joining the rotating point with the center of rotation), or basis set, or coordinate system.

[3] In kinematics, linear means “along a straight or curved line” (the path of the particle in space). In mathematics, however, linear has a different meaning. In both contexts, the word “linear” is related to the word “line”. In mathematics, a line is often defined as a straight curve. For those who adopt this definition, a curve can be straight, and curved lines are not supposed to exist. In kinematics, the term line is used as a synonym of the term trajectory, or path (namely, it has the same non-restricted meaning as that given, in mathematics, to the word curve). In short, both straight and curved lines are supposed to exist. In kinematics and dynamics, the following words refer to the same non-restricted meaning of the term “line":

• “linear” (= along a straight or curved line), • “rectilinear” (= along a straight line, from Latin rectus = straight, and linere = spread), • “curvilinear” (=along a curved line, from Latin curvus = curved, and linere = spread).

In topology and meteorology, the term “line” has the same meaning; namely, a contour line is a curve.

[4] Kane, Thomas; Levinson, David (1996). “2-4 Auxiliary Reference Frames”. Dynamics Online. Sunnyvale, California: OnLine Dynamics, Inc.

[5] Kane, Thomas; Levinson, David (1996). “2-6 Velocity and Acceleration”. Dynamics Online. Sunnyvale, California: OnLine Dynamics, Inc.

[6] Kane, Thomas; Levinson, David (1996). “2-7 Two Points Fixed on a Rigid Body”. Dynamics Online. Sunnyvale, California: OnLine Dynamics, Inc.

[7] Kane, Thomas; Levinson, David (1996). “2-8 One Point Moving on a Rigid Body”. Dynamics Online. Sunnyvale, Califor- nia: OnLine Dynamics, Inc. 220 CHAPTER 33. RIGID BODY

33.8 References

• Roy Featherstone (1987). Robot Dynamics Algorithms. Springer. ISBN 0-89838-230-0. This reference ef- fectively combines screw theory with rigid body dynamics for robotic applications. The author also chooses to use spatial accelerations extensively in place of material accelerations as they simplify the equations and allow for compact notation. • JPL DARTS page has a section on spatial operator algebra (link: ) as well as an extensive list of references (link: ). Chapter 34

Rotation

This article is about movement of a physical body. For other uses, see Rotation (disambiguation). A rotation is a circular movement of an object around a center (or point) of rotation . A three-dimensional object always rotates around an imaginary line called a rotation axis. If the axis passes through the body’s center of mass, the body is said to rotate upon itself, or spin. A rotation about an external point, e.g. the Earth about the Sun, is called a revolution or orbital revolution, typically when it is produced by gravity.

34.1 Mathematics

Main article: Rotation (mathematics) Mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. This definition applies to rotations within both two and three dimensions (in a plane and in space, respectively.) All rigid body movements are rotations, translations, or combinations of the two. A rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion. The axis is 90 degrees perpendicular to the plane of the motion. If the axis of the rotation lies external of the body in question then the body is said to orbit. There is no fundamental difference between a “rotation” and an “orbit” and or “spin”. The key distinction is simply where the axis of the rotation lies, either within or outside of a body in question. This distinction can be demonstrated for both “rigid” and “non rigid” bodies. If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results. The reverse (inverse) of a rotation is also a rotation. Thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, e.g. a translation. Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis, and followed by a rotation around the z axis. That is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the principal rotations are known as yaw, pitch, and roll (known as Tait-Bryan angles). This terminology is also used in computer graphics. See also: curl (mathematics), cyclic permutation, Euler angles, rigid body, rotation around a fixed axis, rotation group SO(3), rotation matrix, axis angle, quaternion and isometry

34.2 Astronomy

In astronomy, rotation is a commonly observed phenomenon. Stars, planets and similar bodies all spin around on their axes. The rotation rate of planets in the solar system was first measured by tracking visual features. Stellar rotation is measured through Doppler shift or by tracking active surface features. This rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect

221 222 CHAPTER 34. ROTATION

A sphere rotating about an axis

of gravity the closer one is to the equator. One effect is that an object weighs slightly less at the equator. Another is that the Earth is slightly deformed into an oblate spheroid. Another consequence of the rotation of a planet is the phenomenon of precession. Like a gyroscope, the overall effect is a slight “wobble” in the movement of the axis of a planet. Currently the tilt of the Earth's axis to its orbital plane (obliquity of the ecliptic) is 23.44 degrees, but this angle changes slowly (over thousands of years). (See also Precession of the equinoxes and Pole star.) 34.2. ASTRONOMY 223

Rotation of a planar figure around a point

34.2.1 Rotation and revolution

Main article: Orbital revolution

While revolution is often used as a synonym for rotation, in many fields, particularly astronomy and related fields, revolution, often referred to as orbital revolution for clarity, is used when one body moves around another while rotation is used to mean the movement around an axis. Moons revolve around their planet, planets revolve about their star (such as the Earth around the Sun); and stars slowly revolve about their galaxial center. The motion of the components of galaxies is complex, but it usually includes a rotation component.

34.2.2 Retrograde rotation

Main article: Retrograde motion

Most planets in our solar system, including Earth, spin in the same direction as they orbit the Sun. The exceptions are 224 CHAPTER 34. ROTATION

Rotational Orbit v Spin

Venus and Uranus. Uranus rotates nearly on its side relative to its orbit. Current speculation is that Uranus started off with a typical prograde orientation and was knocked on its side by a large impact early in its history. Venus may be thought of as rotating slowly backwards (or being “upside down”). The dwarf planet Pluto (formerly considered a planet) is anomalous in this and other ways.

34.3 Physics

Further information: Angular momentum

The speed of rotation is given by the angular frequency (rad/s) or frequency (turn/s, turns per minute), or period (seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s²), This change is caused by torque. The ratio of the two (how heavy is it to start, stop, or otherwise change rotation) is given by the moment of inertia. The angular velocity vector (an axial vector) also describes the direction of the axis of rotation. Similarly the torque is an axial vector. The physics of the rotation around a fixed axis is mathematically described with the axis–angle representation of rotations. According to the right-hand rule, the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation, like a screw. See also: Speed § Tangential Speed, rotational energy, angular velocity, Centrifugal force (fictitious), centripetal force, circular motion, circular orbit, Coriolis effect, spin (physics), rotational spectroscopy and Rigid body dynamics § Linear and angular momentum 34.3. PHYSICS 225

Relations between rotation axis, plane of orbit and axial tilt (for Earth).

Star trails caused by the Earth’s rotation during the camera’s long exposure time.[1] 226 CHAPTER 34. ROTATION

34.3.1 Cosmological principle

The laws of physics are currently believed to be invariant under any fixed rotation. (Although they do appear to change when viewed from a rotating viewpoint: see rotating frame of reference.) In modern physical cosmology, the cosmological principle is the notion that the distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throughout the universe and have no preferred direction, and should, therefore, produce no observable irregularities in the large scale structuring over the course of evolution of the matter field that was initially laid down by the Big Bang. In particular, for a system which behaves the same regardless of how it is oriented in space, its Lagrangian is rotation- ally invariant. According to Noether’s theorem, if the action (the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved.

34.3.2 Euler rotations

Main article: Euler angles Euler rotations provide an alternative description of a rotation. It is a composition of three rotations defined as the movement obtained by changing one of the Euler angles while leaving the other two constant. Euler rotations are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. They constitute a mixed axes of rotation system, where the first angle moves the line of nodes around the external axis z, the second rotates around the line of nodes and the third one is an intrinsic rotation around an axis fixed in the body that moves. These rotations are called precession, nutation, and intrinsic rotation.

34.4 Flight dynamics

Main article: Aircraft principal axes In flight dynamics, the principal rotations described with Euler angles above are known as pitch, roll and yaw. The term rotation is also used in aviation to refer to the upward pitch (nose moves up) of an aircraft, particularly when starting the climb after takeoff. Principal rotations have the advantage of modelling a number of physical systems such as gimbals, and joysticks, so are easily visualised, and are a very compact way of storing a rotation. But they are difficult to use in calculations as even simple operations like combining rotations are expensive to do, and suffer from a form of gimbal lock where the angles cannot be uniquely calculated for certain rotations.

34.5 Amusement rides

Many amusement rides provide rotation. A Ferris wheel has a horizontal central axis, and parallel axes for each gondola, where the rotation is opposite, by gravity or mechanically. As a result, at any time the orientation of the gondola is upright (not rotated), just translated. The tip of the translation vector describes a circle. A carousel provides rotation about a vertical axis. Many rides provide a combination of rotations about several axes. In Chair-O-Planes the rotation about the vertical axis is provided mechanically, while the rotation about the horizontal axis is due to the centripetal force. In roller coaster inversions the rotation about the horizontal axis is one or more full cycles, where inertia keeps people in their seats.

34.6 Sports

“Spin move” redirects here. For other uses, see Spin move (disambiguation).

Rotation of a ball or other object, usually called spin, plays a role in many sports, including topspin and backspin in tennis, English, follow and draw in billiards and pool, curve balls in baseball, spin bowling in cricket, flying disc 34.6. SPORTS 227

P N

R

Euler rotations of the Earth. Intrinsic (green), Precession (blue) and Nutation (red)

sports, etc. Table tennis paddles are manufactured with different surface characteristics to allow the player to impart a greater or lesser amount of spin to the ball. Rotation of a player one or more times around a vertical axis may be called spin in figure skating, twirling (of the baton or the performer) in baton twirling, or 360, 540, 720, etc. in snowboarding, etc. Rotation of a player or performer one or more times around a horizontal axis may be called a flip, roll, somersault, heli, etc. in gymnastics, waterskiing, or many other sports, or a one-and-a-half, two-and-a-half, gainer (starting facing away from the water), etc. in diving, etc. A combination of vertical and horizontal rotation (back flip with 360°) is called a möbius in waterskiing freestyle jumping. Rotation of a player around a horizontal axis, generally between 180 and 360 degrees, may be called a spin move 228 CHAPTER 34. ROTATION

The principal axes of rotation in space and is used as a deceptive or avoidance maneuver, or in an attempt to play, pass, or receive a ball or puck, etc., or to afford a player a view of the goal or other players. It is often seen in hockey, basketball, football of various codes, tennis, etc.

34.7 See also

• Absolute rotation • Attitude (geometry) • Balancing machine • Circular motion • Mach’s principle • Rolling • Rotation around a fixed axis • Rotation formalisms in three dimensions • Rotating locomotion in living systems • Truck bolster

34.8 References

[1] “An Oasis, or a Secret Lair?". ESO Picture of the Week. Retrieved 8 October 2013. 34.9. EXTERNAL LINKS 229

34.9 External links

• Hazewinkel, Michiel, ed. (2001), “Rotation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608- 010-4 • Product of Rotations at cut-the-knot. cut-the-knot.org

• When a Triangle is Equilateral at cut-the-knot. cut-the-knot.org

• Rotate Points Using Polar Coordinates, howtoproperly.com • Rotation in Two Dimensions by Sergio Hannibal Mejia after work by Roger Germundsson and Understanding 3D Rotation by Roger Germundsson, Wolfram Demonstrations Project. demonstrations.wolfram.com Chapter 35

Scaling (geometry)

In Euclidean geometry, uniform scaling (or isotropic scaling[1]) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions. The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc. More general is scaling with a separate scale factor for each axis direction. Non-uniform scaling (anisotropic scaling) is obtained when at least one of the scaling factors is different from the others; a special case is directional scaling or stretching (in one direction). Non-uniform scaling changes the shape of the object; e.g. a square may change into a rectangle, or into a parallelogram if the sides of the square are not parallel to the scaling axes (the angles between lines parallel to the axes are preserved, but not all angles). It occurs, for example, when a faraway billboard is viewed from an oblique angle, or when the shadow of a flat object falls on a surface that is not parallel to it. When the scale factor is larger than 1, (uniform or non-uniform) scaling is sometimes also called dilation or en- largement. When the scale factor is a positive number smaller than 1, scaling is sometimes also called contraction. In the most general sense, a scaling includes the case in which the directions of scaling are not perpendicular. It also includes the case in which one or more scale factors are equal to zero (projection), and the case of one or more negative scale factors (a directional scaling by −1 is equivalent to a reflection). Scaling is a linear transformation, and a special case of homothetic transformation. In most cases, the homothetic transformations are non-linear transformations.

35.1 Matrix representation

A scaling can be represented by a scaling matrix. To scale an object by a vector v = (vx, vy, vz), each point p = (px, py, pz) would need to be multiplied with this scaling matrix:

  vx 0 0   Sv = 0 vy 0 . 0 0 vz As shown below, the multiplication will give the expected result:

     vx 0 0 px vxpx      Svp = 0 vy 0 py = vypy . 0 0 vz pz vzpz Such a scaling changes the diameter of an object by a factor between the scale factors, the area by a factor between the smallest and the largest product of two scale factors, and the volume by the product of all three. The scaling is uniform if and only if the scaling factors are equal (vx = vy = vz). If all except one of the scale factors are equal to 1, we have directional scaling.

230 35.2. USING HOMOGENEOUS COORDINATES 231

In the case where vx = vy = vz = k, scaling increases the area of any surface by a factor of k2 and the volume of any solid object by a factor of k3.

35.1.1 Scaling in arbitrary dimensions

In n -dimensional space Rn , uniform scaling by a factor v is accomplished by scalar multiplication with v , that is, multiplying each coordinate of each point by v . As a special case of linear transformation, it can be achieved also by multiplying each point (viewed as a column vector) with a diagonal matrix whose entries on the diagonal are all equal to v , namely vI . Non-uniform scaling is accomplished by multiplication with any symmetric matrix. The eigenvalues of the matrix are the scale factors, and the corresponding eigenvectors are the axes along which each scale factor applies. A special case is a diagonal matrix, with arbitrary numbers v1, v2, . . . vn along the diagonal: the axes of scaling are then the coordinate axes, and the transformation scales along each axis i by the factor vi In uniform scaling with a non-zero scale factor, all non-zero vectors retain their direction (as seen from the origin), or all have the direction reversed, depending on the sign of the scaling factor. In non-uniform scaling only the vectors that belong to an eigenspace will retain their direction. A vector that is the sum of two or more non-zero vectors belonging to different eigenspaces will be tilted towards the eigenspace with largest eigenvalue.

35.2 Using homogeneous coordinates

In projective geometry, often used in computer graphics, points are represented using homogeneous coordinates. To scale an object by a vector v = (vx, vy, vz), each homogeneous coordinate vector p = (px, py, pz, 1) would need to be multiplied with this projective transformation matrix:

  vx 0 0 0    0 vy 0 0 Sv =  . 0 0 vz 0 0 0 0 1

As shown below, the multiplication will give the expected result:

     vx 0 0 0 px vxpx       0 vy 0 0py vypy  Svp =    =  . 0 0 vz 0 pz vzpz 0 0 0 1 1 1

Since the last component of a homogeneous coordinate can be viewed as the denominator of the other three compo- nents, a uniform scaling by a common factor s (uniform scaling) can be accomplished by using this scaling matrix:

  1 0 0 0 0 1 0 0 S =  . v 0 0 1 0 1 0 0 0 s For each vector p = (px, py, pz, 1) we would have

     1 0 0 0 px px      0 1 0 0py py Svp =    =   0 0 1 0 pz pz 1 1 0 0 0 s 1 s which would be homogenized to 232 CHAPTER 35. SCALING (GEOMETRY)

  spx   spy  . spz 1

35.3 Footnotes

[1] Durand; Cutler. “Transformations” (PowerPoint). Massachusetts Institute of Technology. Retrieved 12 September 2008.

35.4 See also

• Scale (ratio)

• Scale (map) • Scales of scale models

• Scale (disambiguation) • Scaling in gravity

• Transformation matrix

• 3D Scaling

35.5 External links

• Understanding 2D Scaling and Understanding 3D Scaling by Roger Germundsson, The Wolfram Demonstra- tions Project. Chapter 36

Semidirect product

In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: an inner semidirect product is a particular way in which a group can be constructed from two subgroups, one of which is a normal subgroup, while an outer semidirect product is a cartesian product as a set, but with a particular multiplication operation. As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as semidirect products. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (aka split[ting] extension).

36.1 Some equivalent definitions of inner semidirect products

Let G be a group with identity element e, a subgroup H and a normal subgroup N (i.e., N ◁ G). With this premise, the following statements are equivalent:

• G = NH and N ∩ H = {e}.

• Every element of G can be written in a unique way as a product nh, with n ∈ N and h ∈ H.

• Every element of G can be written in a unique way as a product hn, with h ∈ H and n ∈ N.

• The natural embedding H → G, composed with the natural projection G → G / N, yields an isomorphism between H and the quotient group G / N.

• There exists a homomorphism G → H that is the identity on H and whose kernel is N.

If one (and therefore all) of these statements hold, we say that G is a semidirect product of N and H, written

G = N ⋊ H, or that G splits over N; one also says that G is a semidirect product of H acting on N, or even a semidirect product of H and N. To avoid ambiguity, it is advisable to specify which of the two subgroups is normal.

36.2 Outer semidirect products

Let G be a semidirect product of the normal subgroup N and the subgroup H. Let Aut(N) denote the group of all automorphisms of N. The map φ : H → Aut(N) defined by φ(h) = φh, where φh(n) = hnh−1 for all h in H and n in N, is a group homomorphism. (Note that hnh−1∈N since N is normal in G.) Together N, H and φ determine G up to isomorphism, as we show now.

233 234 CHAPTER 36. SEMIDIRECT PRODUCT

Given any two groups N and H (not necessarily subgroups of a given group) and a group homomorphism φ : H → Aut(N), we can construct a new group N ⋊φ H , called the (outer) semidirect product of N and H with respect to φ, defined as follows.[1]

• As a set, N ⋊φ H is the cartesian product N × H.

• Multiplication of elements in N ⋊φ H is determined by the homomorphism φ . The operation is

∗:(N × H) × (N × H) → N ⋊φ H

defined by

∗ (n1, h1) (n2, h2) = (n1φh1 (n2), h1h2)

for n1, n2 in N and h1, h2 in H.

This defines a group in which the identity element is (eN, eH) and the inverse of the element (n, h) is (φh₋₁(n−1), h−1). Pairs (n,eH) form a normal subgroup isomorphic to N, while pairs (eN, h) form a subgroup isomorphic to H. The full group is a semidirect product of those two subgroups in the sense given earlier. Conversely, suppose that we are given a group G with a normal subgroup N and a subgroup H, such that every element g of G may be written uniquely in the form g = nh where n lies in N and h lies in H. Let φ : H → Aut(N) be the homomorphism given by φ(h) = φh, where

−1 φh(n) = hnh for all n ∈ N,h ∈ H. Then G is isomorphic to the semidirect product N ⋊φ H ; the isomorphism sends the product nh to the tuple (n,h). In G, we have

−1 (n1h1)(n2h2) = n1h1n2h1 h1h2 = (n1φh1 (n2))(h1h2)

which shows that the above map is indeed an isomorphism and also explains the definition of the multiplication rule in N ⋊φ H . The direct product is a special case of the semidirect product. To see this, let φ be the trivial homomorphism, i.e. sending every element of H to the identity automorphism of N, then N ⋊φ H is the direct product N × H . A version of the splitting lemma for groups states that a group G is isomorphic to a semidirect product of the two groups N and H if and only if there exists a short exact sequence

1 −→ N −→β G −→α H −→ 1

and a group homomorphism γ : H → G such that α ∘ γ = idH, the identity map on H. In this case, φ:H → Aut(N) is given by φ(h) = φh, where

−1 −1 φh(n) = β (γ(h)β(n)γ(h )).

36.3 Examples

[2] The dihedral group D₂n with 2n elements is isomorphic to a semidirect product of the cyclic groups Cn and C2. Here, the non-identity element of C2 acts on Cn by inverting elements; this is an automorphism since Cn is abelian. The presentation for this group is: 36.4. PROPERTIES 235

⟨a, b | a2 = e, bn = e, aba−1 = b−1⟩.

More generally, a semidirect product of any two cyclic groups Cm with generator a and Cn with generator b is given by a single relation aba−1 = bk with k and n coprime, i.e. the presentation:[2]

⟨a, b | am = e, bn = e, aba−1 = bk ⟩.

r r −r kr If r and m are coprime, a is a generator of Cm and a ba = b , hence the presentation:

r ⟨a, b | am = e, bn = e, aba−1 = bk ⟩

gives a group isomorphic to the previous one. The fundamental group of the Klein bottle can be presented in the form

⟨a, b | aba−1 = b−1 ⟩

and is therefore a semidirect product of the group of integers, Z , with Z . The corresponding homomorphism φ : Z → Aut(Z) is given by φ(h)(n) = (−1)hn . The Euclidean group of all rigid motions (isometries) of the plane (maps f : R2 → R2 such that the Euclidean distance between x and y equals the distance between f(x) and f(y) for all x and y in R2) is isomorphic to a semidirect product of the abelian group R2 (which describes translations) and the group O(2) of orthogonal 2×2 matrices (which describes rotations and reflections that keep the origin fixed). Applying a translation and then a rotation or reflection has the same effect as applying the rotation or reflection first and then a translation by the rotated or reflected translation vector (i.e. applying the conjugate of the original translation). This shows that the group of translations is a normal subgroup of the Euclidean group, that the Euclidean group is a semidirect product of the translation group and O(2), and that the corresponding homomorphism φ : O(2) → Aut(R2) is given by matrix multiplication: φ(h)(n) = hn . The orthogonal group O(n) of all orthogonal real n×n matrices (intuitively the set of all rotations and reflections of n- dimensional space which keep the origin fixed) is isomorphic to a semidirect product of the group SO(n) (consisting of all orthogonal matrices with determinant 1, intuitively the rotations of n-dimensional space) and C2. If we represent C2 as the multiplicative group of matrices {I, R}, where R is a reflection of n dimensional space which keeps the origin fixed (i.e. an orthogonal matrix with determinant –1 representing an involution), then φ : C2 → Aut(SO(n)) is −1 given by φ(H)(N) = HNH for all H in C2 and N in SO(n). In the non-trivial case ( H is not the identity) this means that φ(H) is conjugation of operations by the reflection (a rotation axis and the direction of rotation are replaced by their “mirror image”). The group of semilinear transformations on a vector space V over a field K , often denoted L(V ) , is isomorphic to a semidirect product of the linear group GL(V ) (a normal subgroup of L(V ) ), and the automorphism group of K .

36.4 Properties

As a consequence of Lagrange’s theorem, if G is the semidirect product of the normal subgroup N and the subgroup H, and both N and H are finite, then the order of G equals the product of the orders of N and H.

36.4.1 Relation to direct products

Suppose G is a semidirect product of the normal subgroup N and the subgroup H. If H is also normal in G, or equivalently, if there exists a homomorphism G → N which is the identity on N, then G is the direct product of N and H. The direct product of two groups N and H can be thought of as the semidirect product of N and H with respect to φ(h) = idN for all h in H. 236 CHAPTER 36. SEMIDIRECT PRODUCT

Note that in a direct product, the order of the factors is not important, since N × H is isomorphic to H × N. This is not the case for semidirect products, as the two factors play different roles. Furthermore, the result of a (proper) semidirect product by means of a non-trivial homomorphism is never an abelian group, even if the factor groups are abelian.

36.4.2 Non-uniqueness of semidirect products (and further examples)

As opposed to the case with the direct product, a semidirect product of two groups is not, in general, unique; if G and G′ are two groups which both contain isomorphic copies of N as a normal subgroup and H as a subgroup, and both are a semidirect product of N and H, then it does not follow that G and G′ are isomorphic because the semidirect product also depends on the choice of an action of H on N.

For example, there are four non-isomorphic groups of order 16 that are [semi]direct product of C8 and C2; C8 is necessarily a normal subgroup in this case because it has index 2 in a group of order 16. One of these four [semi]direct products is the direct product, while the other three are non-abelian groups:

• the dihedral group of order 16

• the quasidihedral group of order 16

• the Iwasawa group of order 16 (M16)

For comparative Cayley diagrams of these see Example 3 in . If a given group is a semidirect product, then there is no guarantee that this decomposition is unique. For example, there is a group of order 24 (the only one containing six elements of order 4 and six elements of order 6) that can be [3] expressed as semidirect product in the following ways: D8 ⋉ C3 ≃ C2 ⋉ Q12 ≃ C2 ⋉ D12 ≃ D6 ⋉ V.

36.4.3 Existence

Main article: Schur–Zassenhaus theorem

In general, there’s no known characterization (necessary and sufficient condition) for the existence of semidirect products in groups. However, some sufficient conditions are known, which guarantee existence in certain cases. For finite groups, the Schur–Zassenhaus theorem guarantees existence of a semidirect product when the order of the normal subgroup is coprime to the order of the quotient group. For example, the Schur–Zassenhaus theorem implies the existence of a semi-direct product among groups of order 6; there are two such products, one of which is a direct product, and the other a dihedral group. In contrast, the Schur–Zassenhaus theorem does not say anything about groups of order 4 or groups of order 8 for instance.

36.4.4 When are they distinct?

36.5 Generalizations

The construction of semidirect products can be pushed much further. The Zappa–Szep product of groups is a gener- alization which, in its internal version, does not assume that either subgroup is normal. There is also a construction in ring theory, the crossed product of rings. This is seen naturally as soon as one constructs a group ring for a semidirect product of groups. There is also the semidirect sum of Lie algebras. Given a group action on a topological space, there is a corresponding crossed product which will in general be non-commutative even if the group is abelian. This kind of ring (see crossed product for a related construction) can play the role of the space of orbits of the group action, in cases where that space cannot be approached by conventional topological techniques – for example in the work of Alain Connes (cf. noncommutative geometry). There are also far-reaching generalisations in category theory. They show how to construct fibred categories from indexed categories. This is an abstract form of the outer semidirect product construction. 36.6. NOTATION 237

36.5.1 Groupoids

Another generalization is for groupoids. This occurs in topology because if a group G acts on a space X it also acts on the fundamental groupoid π1(X) of the space. The semidirect product π1(X) ⋊ G is then relevant to finding the fundamental groupoid of the orbit space X/G . For full details see Chapter 11 of the book referenced below, and also some details in semidirect product[4] in ncatlab.

36.5.2 Abelian categories

Non-trivial semidirect products do not arise in abelian categories, such as the category of modules. In this case, the splitting lemma shows that every semidirect product is a direct product. Thus the existence of semidirect products reflects a failure of the category to be abelian.

36.6 Notation

Usually the semidirect product of a group H acting on a group N (in most cases by conjugation as subgroups of a common group) is denoted by N ⋊ H or H ⋉ N . However, some sources may use this symbol with the opposite meaning. In case the action ϕ : H → Aut(N) should be made explicit, one also writes N ⋊ϕ H . One way of thinking about the N ⋊ H symbol is as a combination of the symbol for normal subgroup ( ◁ ) and the symbol for the product ( × ). Unicode lists four variants:[5]

Here the Unicode description of the rtimes symbol says “right normal factor”, in contrast to its usual meaning in mathematical practice. In LaTeX, the commands \rtimes and \ltimes produce the corresponding characters.

36.7 See also

• Lie algebra semidirect sum

• Holomorph

• Subdirect product

• Wreath product

• Affine Lie algebra

36.8 Notes

[1] Robinson, Derek John Scott (2003). An Introduction to Abstract Algebra. Walter de Gruyter. pp. 75–76. ISBN 9783110175448.

[2] Mac Lane, Saunders; Birkhoff, Garrett (1999). Algebra (3 ed.). American Mathematical Society. pp. 414–415. ISBN 0-8218-1646-2.

[3] H.E. Rose (2009). A Course on Finite Groups. Springer Science & Business Media. p. 183. ISBN 978-1-84882-889-6. Note that Rose uses the opposite notation convention than the one adopted on this page (p. 152).

[4] Ncatlab.org

[5] See unicode.org 238 CHAPTER 36. SEMIDIRECT PRODUCT

36.9 References

• R. Brown, Topology and groupoids, Booksurge 2006. ISBN 1-4196-2722-8 Chapter 37

Shear mapping

A horizontal shearing of the plane with coefficient m = 1.25, illustrated by its effect (in green) on a rectangular grid and some figures (in blue). The black dot is the origin.

In plane geometry, a shear mapping is a linear map that displaces each point in fixed direction, by an amount proportional to its signed distance from a line that is parallel to that direction.[1] This type of mapping is also called shear transformation, transvection, or just shearing. An example is the mapping that takes any point with coordinates (x, y) to the point (x + 2y, y) . In this case, the displacement is horizontal, the fixed line is the x -axis, and the signed distance is the y coordinate. Note that points on opposite sides of the reference line are displaced in opposite directions. Shear mappings must not be confused with rotations. Applying a shear map to a set of points of the plane will change all angles between them (except straight angles), and the length of any line segment that is not parallel to the direction of displacement. Therefore it will usually distort the shape of a geometric figure, for example turning squares into non-square parallelograms, and circles into ellipses. However a shearing does preserve the area of geometric figures, the alignment and relative distances of collinear points. A shear mapping is the main difference between the upright and slanted (or italic) styles of letters. The same definition is used in three-dimensional geometry, except that the distance is measured from a fixed plane. A three-dimensional shearing transformation preserves the volume of solid figures, but changes areas of plane figures (except those that are parallel to the displacement). This transformation is used to describe laminar flow of a fluid between plates, one moving in a plane above and parallel to the first. In the general n -dimensional Cartesian space Rn , the distance is measured from a fixed hyperplane parallel to the direction of displacement. This geometric transformation is a linear transformation of Rn that preserves the n -dimensional measure (hypervolume) of any set.

239 240 CHAPTER 37. SHEAR MAPPING

y dimension boundary plate (2D, moving) velocity, u

fluid gradient,

boundary plate (2D, stationary)

In fluid dynamics a shear mapping depicts fluid flow between parallel plates in relative motion.

37.1 Definition

37.1.1 Horizontal and vertical shear of the plane

In the plane R2 = R × R , a horizontal shear (or shear parallel to the x axis) is a function that takes a generic point with coordinates (x, y) to the point (x + my, y) ; where m is a fixed parameter, called the shear factor. The effect of this mapping is to displace every point horizontally by an amount proportionally to its y coordinate. Any point above the x -axis is displaced to the right (increasing x ) if m > 0 , and to the left if m < 0 . Points below the x -axis move in the opposite direction, while points on the axis stay fixed. Straight lines parallel to the x -axis remain where they are, while all other lines are turned, by various angles, about the point where they cross the x -axis. Vertical lines, in particular, become oblique lines with slope 1/m . Therefore the shear factor m is the cotangent of the angle φ by which the vertical lines tilt, called the shear angle. If the coordinates of a point are written as a column vector (a 2×1 matrix), the shear mapping can be written as multiplication by a 2×2 matrix:

( ) ( ) ( )( ) x′ x + my 1 m x = = . y′ y 0 1 y A vertical shear (or shear parallel to the y -axis) of lines is similar, except that the roles of x and y are swapped. It corresponds to multiplying the coordinate vector by the transposed matrix:

( ) ( ) ( )( ) x′ x 1 0 x = = . y′ mx + y m 1 y The vertical shear displaces points to the right of the y -axis up or down, depending on the sign of m . It leaves vertical lines invariant, but tilts all other lines about the point where they meet the y -axis. Horizontal lines, in particular, get tilted by the shear angle φ to become lines with slope m . 37.1. DEFINITION 241

transform= "skewX(-30)"

Through a shear mapping coded in SVG, a rectangle becomes a rhombus.

37.1.2 General shear mappings

For a vector space V and subspace W, a shear fixing W translates all vectors parallel to W. To be more precise, if V is the direct sum of W and W′, and we write vectors as

v = w + w′

correspondingly, the typical shear fixing W is L where

L(v) = (w + Mw′) + w ′

where M is a linear mapping from W′ into W. Therefore in block matrix terms L can be represented as

( ) IM 0 I 242 CHAPTER 37. SHEAR MAPPING

37.2 Applications

The following applications of shear mapping were noted by William Kingdon Clifford:

“A succession of shears will enable us to reduce any figure bounded by straight lines to a triangle of equal area.” "... we may shear any triangle into a right-angled triangle, and this will not alter its area. Thus the area of any triangle is half the area of the rectangle on the same base and with height equal to the perpendicular on the base from the opposite angle.”[2]

The area-preserving property of a shear mapping can be used for results involving area. For instance, the Pythagorean theorem has been illustrated with shear mapping.[3] An algorithm due to Alan W. Paeth uses a sequence of three shear mappings (horizontal, vertical, then horizontal again) to rotate a digital image by an arbitrary angle. The algorithm is very simple to implement, and very efficient, since each step processes only one column or one row of pixels at a time.[4]

37.3 References

[1] Definition according to Weisstein, Eric W. Shear From MathWorld − A Wolfram Web Resource

[2] William Kingdon Clifford (1885) Common Sense and the Exact Sciences, page 113

[3] Mike May S.J. Pythagorean theorem by shear mapping, from Saint Louis University; requires Java and Geogebra. Click on the “Steps” slider and observe shears at steps 5 and 6.

[4] Alan Paeth (1986), A Fast Algorithm for General Raster Rotation. Proceedings of Graphics Interface '86, pages 77–81. Chapter 38

Similarity (geometry)

For other uses, see Similarity transformation and Similarity (disambiguation). Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the

Figures shown in the same color are similar mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or shrinking), possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other when zoomed in or out at some level. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure.

243 244 CHAPTER 38. SIMILARITY (GEOMETRY)

This article assumes that a scaling can have a scale factor of 1, so that all congruent shapes are also similar, but some school text books specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar.

38.1 Similar triangles

In geometry two triangles, △ABC and △A′B′C′ , are similar if and only if corresponding angles are congruent and the lengths of corresponding sides are proportional.[1] It can be shown that two triangles having congruent angles (equiangular triangles) are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem.[2] Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.[3] There are several statements each of which is necessary and sufficient for two triangles to be similar: 1. The triangles have two congruent angles,[4] which in Euclidean geometry implies that all their angles are congruent.[5] That is:

If ∠BAC is equal in measure to ∠B′A′C′ , and ∠ABC is equal in measure to ∠A′B′C′ , then this implies that ∠ACB is equal in measure to ∠A′C′B′ and the triangles are similar.

2. All the corresponding sides have lengths in the same ratio:[6]

AB BC AC A′B′ = B′C′ = A′C′ . This is equivalent to saying that one triangle (or its mirror image) is an enlargement of the other.

3. Two sides have lengths in the same ratio, and the angles included between these sides have the same measure.[7] For instance:

AB BC ∠ ∠ ′ ′ ′ A′B′ = B′C′ and ABC is equal in measure to A B C .

This is known as the SAS Similarity Criterion.[8] When two triangles △ABC and △A′B′C′ are similar, one writes[9]:p. 22

△ABC ∼ △A′B′C′

There are several elementary results concerning similar triangles in Euclidean geometry:[10]

• Any two equilateral triangles are similar.

• Two triangles, both similar to a third triangle, are similar to each other (transitivity of similarity of triangles).

• Corresponding altitudes of similar triangles have the same ratio as the corresponding sides.

• Two right triangles are similar if the hypotenuse and one other side have lengths in the same ratio.[11]

Given a triangle △ABC and a line segment DE one can, with straightedge and compass, find a point F such that △ABC ∼ △DEF . The statement that the point F satisfying this condition exists is Wallis’s Postulate[12] and is logically equivalent to Euclid’s Parallel Postulate.[13] In hyperbolic geometry (where Wallis’s Postulate is false) similar triangles are congruent. In the axiomatic treatment of Euclidean geometry given by G.D. Birkhoff (see Birkhoff’s axioms) the SAS Similarity Criterion given above was used to replace both Euclid’s Parallel Postulate and the SAS axiom which enabled the dramatic shortening of Hilbert’s axioms.[8] 38.2. OTHER SIMILAR POLYGONS 245

38.2 Other similar polygons

The concept of similarity extends to polygons with more than three sides. Given any two similar polygons, corre- sponding sides taken in the same sequence (even if clockwise for one polygon and counterclockwise for the other) are proportional and corresponding angles taken in the same sequence are equal in measure. However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles (otherwise, for exam- ple, all rhombi would be similar). Likewise, equality of all angles in sequence is not sufficient to guarantee similarity (otherwise all rectangles would be similar). A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional.

38.3 Similar curves

Several types of curves have the property that all examples of that type are similar to each other. These include:

• Circles

• Parabolas [14]

• Hyperbolas of a specific eccentricity[15]

• Ellipses of a specific eccentricity[15]

• Catenaries

• Graphs of the logarithm function for different bases

• Graphs of the exponential function for different bases

• Logarithmic spirals

38.4 Similarity in Euclidean space

A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection f from the space onto itself that multiplies all distances by the same positive real number r, so that for any two points x and y we have

d(f(x), f(y)) = rd(x, y), where "d(x,y)" is the Euclidean distance from x to y.[16] The scalar r has many names in the literature including; the ratio of similarity, the stretching factor and the similarity coefficient. When r = 1 a similarity is called an isometry (rigid motion). Two sets are called similar if one is the image of the other under a similarity. Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments.[17] Similarities preserve angles but do not necessarily preserve orientation, direct similitudes preserve ori- entation and opposite similitudes change it.[18] The similarities of Euclidean space form a group under the operation of composition called the similarities group S.[19] The direct similitudes form a normal subgroup of S and the Euclidean group E(n) of isometries also forms a normal subgroup.[20] The similarities group S is itself a subgroup of the affine group, so every similarity is an affine transformation. One can view the Euclidean plane as the complex plane,[21] that is, as a 2-dimensional space over the reals. The 2D similarity transformations can then be expressed in terms of complex arithmetic and are given by f(z) = az + b (direct similitudes) and f(z) = az + b (opposite similitudes) where a and b are complex numbers, a ≠ 0. When |a| = 1, these similarities are isometries. 246 CHAPTER 38. SIMILARITY (GEOMETRY)

38.5 Ratios of sides, of areas, and of volumes

Main article: Square-cube law

The ratio between the areas of similar figures is equal to the square of the ratio of corresponding lengths of those figures (for example, when the side of a square or the radius of a circle is multiplied by three, its area is multiplied by nine — i.e. by three squared). The altitudes of similar triangles are in the same ratio as corresponding sides. If a triangle has a side of length b and an altitude drawn to that side of length h then a similar triangle with corresponding side of length kb will have an altitude drawn to that side of length kh. The area of the first triangle is, A = bh/2, while the area of the similar triangle will be A* = (kb)(kh)/2 = k2A. Similar figures which can be decomposed into similar triangles will have areas related in the same way. The relationship holds for figures that are not rectifiable as well. The ratio between the volumes of similar figures is equal to the cube of the ratio of corresponding lengths of those figures (for example, when the edge of a cube or the radius of a sphere is multiplied by three, its volume is multiplied by 27 — i.e. by three cubed). Galileo’s square–cube law concerns similar solids. If the ratio of similitude (ratio of corresponding sides) between the solids is k, then the ratio of surface areas of the solids will be k2, while the ratio of volumes will be k3.

38.6 Similarity in general metric spaces

Sierpinski triangle. A space having self-similarity dimension ln 3 / ln 2 = log23, which is approximately 1.58. (from Hausdorff dimension.)

In a general metric space (X, d), an exact similitude is a function f from the metric space X into itself that multiplies 38.7. TOPOLOGY 247

all distances by the same positive scalar r, called f’s contraction factor, so that for any two points x and y we have

d(f(x), f(y)) = rd(x, y).

Weaker versions of similarity would for instance have f be a bi-Lipschitz function and the scalar r a limit

d(f(x), f(y)) lim = r. d(x, y)

This weaker version applies when the metric is an effective resistance on a topologically self-similar set.

A self-similar subset of a metric space (X, d) is a set K for which there exists a finite set of similitudes {fs}s∈S with contraction factors 0 ≤ rs < 1 such that K is the unique compact subset of X for which

∪ fs(K) = K. s∈S

These self-similar sets have a self-similar measure µD with dimension D given by the formula

∑ D (rs) = 1 s∈S

which is often (but not always) equal to the set’s Hausdorff dimension and packing dimension. If the overlaps between the fs(K) are “small”, we have the following simple formula for the measure:

D ◦ ◦ · · · ◦ · ··· D µ (fs1 fs2 fsn (K)) = (rs1 rs2 rsn ) .

38.7 Topology

In topology, a metric space can be constructed by defining a similarity instead of a distance. The similarity is a function such that its value is greater when two points are closer (contrary to the distance, which is a measure of dissimilarity: the closer the points, the lesser the distance). The definition of the similarity can vary among authors, depending on which properties are desired. The basic common properties are

1. Positive defined: ∀(a, b),S(a, b) ≥ 0

2. Majored by the similarity of one element on itself (auto-similarity): S(a, b) ≤ S(a, a) and ∀(a, b),S(a, b) = S(a, a) ⇔ a = b

More properties can be invoked, such as reflectivity ( ∀(a, b) S(a, b) = S(b, a) ) or finiteness ( ∀(a, b) S(a, b) < ∞ ). The upper value is often set at 1 (creating a possibility for a probabilistic interpretation of the similitude).

38.8 Self-similarity

Self-similarity means that a pattern is non-trivially similar to itself, e.g., the set {.., 0.5, 0.75, 1, 1.5, 2, 3, 4, 6, 8, 12, ..} of numbers of the form {2i, 3 · 2i} where i ranges over all integers. When this set is plotted on a logarithmic scale it has one-dimensional translational symmetry: adding or subtracting the logarithm of two to the logarithm of one of these numbers produces the logarithm of another of these numbers. In the given set of numbers themselves, this corresponds to a similarity transformation in which the numbers are multiplied or divided by two. 248 CHAPTER 38. SIMILARITY (GEOMETRY)

38.9 See also

• Congruence (geometry)

• Hamming distance (string or sequence similarity)

• Inversive geometry

• Jaccard index

• Proportionality

• Semantic similarity

• Similarity search

• Similarity space on Numerical taxonomy

• Homoeoid (shell of concentric, similar ellipsoids)

• Solution of triangles

38.10 Notes

[1] Sibley 1998, p. 35

[2] Stahl 2003, p. 127. This is also proved in Euclid’s Elements, Book VI, Proposition 4.

[3] For instance, Venema 2006, p. 122 and Henderson & Taimiṇa 2005, p. 123

[4] Euclid’s elements Book VI Proposition 4.

[5] This statement is not true in Non-euclidean geometry where the triangle angle sum is not 180 degrees.

[6] Euclid’s elements Book VI Proposition 5

[7] Euclid’s elements Book VI Proposition 6

[8] Venema 2006, p. 143

[9] Posamentier, Alfred S. and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.

[10] Jacobs 1974, pp. 384 - 393

[11] Hadamard, Jacques (2008), Lessons in Geometry, Vol. I: Plane Geometry, American Mathematical Society, Theorem 120, p. 125, ISBN 9780821843673.

[12] Named for John Wallis (1616-1703)

[13] Venema 2006, p. 122

[14] a proof from academia.edu

[15] The shape of an ellipse depends only on the ratio b/a

[16] Smart 1998, p. 92

[17] Yale 1968, p. 47 Theorem 2.1

[18] Pedoe 1988, pp. 179-181

[19] Yale 1968, p. 46

[20] Pedoe 1988, p. 182

[21] This traditional term, as explained in its article, is a misnomer. This is actually the 1-dimensional complex line. 38.11. REFERENCES 249

38.11 References

• Henderson, David W.; Taimina, Daina (2005), Experiencing Geometry/Euclidean and Non-Euclidean with His- tory (3rd ed.), Pearson Prentice-Hall, ISBN 978-0-13-143748-7 • Jacobs, Harold R. (1974), Geometry, W.H. Freeman and Co., ISBN 0-7167-0456-0

• Pedoe, Dan (1988) [1970], Geometry/A Comprehensive Course, Dover, ISBN 0-486-65812-0

• Sibley, Thomas Q. (1998), The Geometric Viewpoint/A Survey of Geometries, Addison-Wesley, ISBN 978-0- 201-87450-1

• Smart, James R. (1998), Modern Geometries (5th ed.), Brooks/Cole, ISBN 0-534-35188-3 • Stahl, Saul (2003), Geometry/From Euclid to Knots, Prentice-Hall, ISBN 978-0-13-032927-1

• Venema, Gerard A. (2006), Foundations of Geometry, Pearson Prentice-Hall, ISBN 978-0-13-143700-5 • Yale, Paul B. (1968), Geometry and Symmetry, Holden-Day

38.12 Further reading

• Judith N. Cederberg (1989, 2001) A Course in Modern Geometries, Chapter 3.12 Similarity Transformations, pp. 183–9, Springer ISBN 0-387-98972-2 .

• H.S.M. Coxeter (1961,9) Introduction to Geometry, §5 Similarity in the Euclidean Plane, pp. 67–76, §7 Isom- etry and Similarity in Euclidean Space, pp 96–104, John Wiley & Sons.

• Günter Ewald (1971) Geometry: An Introduction, pp 106, 181, Wadsworth Publishing. • George E. Martin (1982) Transformation Geometry: An Introduction to Symmetry, Chapter 13 Similarities in the Plane, pp. 136–46, Springer ISBN 0-387-90636-3 .

38.13 External links

• Animated demonstration of similar triangles Chapter 39

Squeeze mapping

r = 3/2 squeeze mapping

In linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping. For a fixed positive real number a, the mapping

(x, y) 7→ (ax, y/a) is the squeeze mapping with parameter a. Since

{(u, v): uv = constant} is a hyperbola, if u = ax and v = y/a, then uv = xy and the points of the image of the squeeze mapping are on the same hyperbola as (x,y) is. For this reason it is natural to think of the squeeze mapping as a hyperbolic rotation, as did Émile Borel in 1914,[1] by analogy with circular rotations which preserve circles.

250 39.1. LOGARITHM AND HYPERBOLIC ANGLE 251

39.1 Logarithm and hyperbolic angle

The squeeze mapping sets the stage for development of the concept of logarithms. The problem of finding the area bounded by a hyperbola (such as xy = 1) is one of quadrature. The solution, found by Grégoire de Saint-Vincent and Alphonse Antonio de Sarasa in 1647, required the natural logarithm function, a new concept. Some insight into logarithms comes through hyperbolic sectors that are permuted by squeeze mappings while preserving their area. The area of a hyperbolic sector is taken as a measure of a hyperbolic angle associated with the sector. The hyperbolic angle concept is quite independent of the ordinary circular angle, but shares a property of invariance with it: whereas circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping. Both circular and hyperbolic angle generate invariant measures but with respect to different transformation groups. The hyperbolic functions, which take hyperbolic angle as argument, perform the role that circular functions play with the circular angle argument.[2]

39.2 Group theory

A squeeze mapping moves one purple hyperbolic sector to another with the same area. It also squeezes blue and green rectangles.

If r and s are positive real numbers, the composition of their squeeze mappings is the squeeze mapping of their product. Therefore the collection of squeeze mappings forms a one-parameter group isomorphic to the multiplicative group of positive real numbers. An additive view of this group arises from consideration of hyperbolic sectors and their hyperbolic angles. 252 CHAPTER 39. SQUEEZE MAPPING

From the point of view of the classical groups, the group of squeeze mappings is SO+(1,1), the identity component of the indefinite orthogonal group of 2 × 2 real matrices preserving the quadratic form u2 − v2. This is equivalent to preserving the form xy via the change of basis

x = u + v, y = u − v , and corresponds geometrically to preserving hyperbolae. The perspective of the group of squeeze mappings as hy- perbolic rotation is analogous to interpreting the group SO(2) (the connected component of the definite orthogonal group) preserving quadratic form x2 + y2) as being circular rotations. Note that the “SO+" notation corresponds to the fact that the reflections

u 7→ −u, v 7→ −v are not allowed, though they preserve the form (in terms of x and y these are x ↦ y, y ↦ x and x ↦ −x, y ↦ −y); the additional "+" in the hyperbolic case (as compared with the circular case) is necessary to specify the identity compo- nent because the group O(1,1) has 4 connected components, while the group O(2) has 2 components: SO(1,1) has 2 components, while SO(2) only has 1. The fact that the squeeze transforms preserve area and orientation corresponds to the inclusion of subgroups SO ⊂ SL – in this case SO(1,1) ⊂ SL(2) – of the subgroup of hyperbolic rotations in the special linear group of transforms preserving area and orientation (a volume form). In the language of Möbius transforms, the squeeze transformations are the hyperbolic elements in the classification of elements.

39.3 Applications

In studying linear algebra there are the purely abstract applications such as illustration of the singular-value decom- position or in the important role of the squeeze mapping in the structure of 2 × 2 real matrices. These applications are somewhat bland compared to two physical and a philosophical application.

39.3.1 Corner flow

In fluid dynamics one of the fundamental motions of an incompressible flow involves bifurcation of a flow running up against an immovable wall. Representing the wall by the axis y = 0 and taking the parameter r = exp(t) where t is time, then the squeeze mapping with parameter r applied to an initial fluid state produces a flow with bifurcation left and right of the axis x = 0. The same model gives fluid convergence when time is run backward. Indeed, the area of any hyperbolic sector is invariant under squeezing. For another approach to a flow with hyperbolic streamlines, see the article potential flow, section “Power law with n = 2”. In 1989 Ottino[3] described the “linear isochoric two-dimensional flow” as

v1 = Gx2 v2 = KGx1 where K lies in the interval [−1, 1]. The streamlines follow the curves

2 − 2 x2 Kx1 = constant so negative K corresponds to an ellipse and positive K to a hyperbola, with the rectangular case of the squeeze mapping corresponding to K = 1. Stocker and Hosoi[4] described their approach to corner flow as follows:

we suggest an alternative formulation to account for the corner-like geometry, based on the use of hy- perbolic coordinates, which allows substantial analytical progress towards determination of the flow in a Plateau border and attached liquid threads. We consider a region of flow forming an angle of π/2 and delimited on the left and bottom by symmetry planes. 39.4. SEE ALSO 253

Stocker and Hosoi then recall Moffatt’s[5] consideration of “flow in a corner between rigid boundaries, induced by an arbitrary disturbance at a large distance.” According to Stocker and Hosoi,

For a free fluid in a square corner, Moffatt’s (antisymmetric) stream function ... [indicates] that hyper- bolic coordinates are indeed the natural choice to describe these flows.

39.3.2 Relativistic spacetime

Select (0,0) for a “here and now” in a spacetime. Light radiant left and right through this central event tracks two lines in the spacetime, lines that can be used to give coordinates to events away from (0,0). Trajectories of lesser velocity track closer to the original timeline (0,t). Any such velocity can be viewed as a zero velocity under a squeeze mapping called a Lorentz boost. This insight follows from a study of split-complex number multiplications and the diagonal basis which corresponds to the pair of light lines. Formally, a squeeze preserves the hyperbolic metric expressed in the form xy; in a different coordinate system. This application in the theory of relativity was noted in 1912 by Wilson and Lewis,[6] by Werner Greub,[7] and by Louis Kauffman.[8] Furthermore, Wolfgang Rindler, in his popular textbook on relativity, used the squeeze mapping form of Lorentz transformations in his demonstration of their characteristic property.[9] The term squeeze transformation was used in this context in an article connecting the Lorentz group with Jones calculus in optics.[10]

39.3.3 Bridge to transcendentals

The area-preserving property of squeeze mapping has an application in setting the foundation of the transcendental functions natural logarithm and its inverse the exponential function: Definition: Sector(a,b) is the hyperbolic sector obtained with central rays to (a, 1/a) and (b, 1/b). Lemma: If bc = ad, then there is a squeeze mapping that moves the sector(a,b) to sector(c,d). Proof: Take parameter r = c/a so that (u,v) = (rx, y/r) takes (a, 1/a) to (c, 1/c) and (b, 1/b) to (d, 1/d). Theorem (Gregoire de Saint-Vincent 1647) If bc = ad, then the quadrature of the hyperbola xy = 1 against the asymptote has equal areas between a and b compared to between c and d. Proof: An argument adding and subtracting triangles of area ½, one triangle being {(0,0), (0,1), (1,1)}, shows the hyperbolic sector area is equal to the area along the asymptote. The theorem then follows from the lemma. Theorem (Alphonse Antonio de Sarasa 1649) As area measured against the asymptote increases in arithmetic pro- gression, the projections upon the asymptote increase in geometric sequence. Thus the areas form logarithms of the asymptote index. For instance, for a standard position angle which runs from (1, 1) to (x, 1/x), one may ask “When is the hyperbolic angle equal to one?" The answer is the transcendental number x = e. A squeeze with r = e moves the unit angle to one between (e, 1/e) and (ee, 1/ee) which subtends a sector also of area one. The geometric progression

e, e2, e3, ..., en, ...

corresponds to the asymptotic index achieved with each sum of areas

1,2,3, ..., n,... which is a proto-typical arithmetic progression A + nd where A = 0 and d = 1 .

39.4 See also

• Equi-areal mapping 254 CHAPTER 39. SQUEEZE MAPPING

• Indefinite orthogonal group

• Isochoric process • Lorentz transformation

39.5 References

[1] Émile Borel (1914) Introduction Geometrique à quelques Théories Physiques, page 29, Gauthier-Villars, link from Cornell University Historical Math Monographs

[2] Mellon W. Haskell (1895) On the introduction of the notion of hyperbolic functions Bulletin of the American Mathematical Society 1(6):155–9,particularly equation 12, page 159

[3] J. M. Ottino (1989) The Kinematics of Mixing: stretching, chaos, transport, page 29, Cambridge University Press

[4] Roman Stocker & A.E. Hosoi (2004) “Corner flow in free liquid films”, Journal of Engineering Mathematics 50:267–88

[5] H.K. Moffatt (1964) “Viscous and resistive eddies near a sharp corner”, Journal of Fluid Mechanics 18:1–18

[6] Edwin Bidwell Wilson & Gilbert N. Lewis (1912) “The space-time manifold of relativity. The non-Euclidean geometry of mechanics and electromagnetics”, Proceedings of the American Academy of Arts and Sciences 48:387–507, footnote p. 401

[7] W. H. Greub (1967) Linear Algebra, Springer-Verlag. See pages 272 to 274

[8] Louis Kauffman (1985) “Transformations in Special Relativity”, International Journal of Theoretical Physics 24:223–36

[9] Wolfgang Rindler, Essential Relativity, equation 29.5 on page 45 of the 1969 edition, or equation 2.17 on page 37 of the 1977 edition, or equation 2.16 on page 52 of the 2001 edition

[10] Daesoo Han, Young Suh Kim & Marilyn E. Noz (1997) “Jones-matrix formalism as a representation of the Lorentz group”, Journal of the Optical Society of America A14(9):2290–8

• HSM Coxeter & SL Greitzer (1967) Geometry Revisited, Chapter 4 Transformations, A genealogy of transfor- mation.

• P. S. Modenov and A. S. Parkhomenko (1965) Geometric Transformations, volume one. See pages 104 to 106. • Walter, Scott (1999). “The non-Euclidean style of Minkowskian relativity” (PDF). In J. Gray. The Symbolic Universe: Geometry and Physics. Oxford University Press. pp. 91–127.(see page 9 of e-link) Chapter 40

Transformation matrix

In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping Rn to Rm and ⃗x is a column vector with n entries, then

T (⃗x) = A⃗x for some m×n matrix A, called the transformation matrix of T. There are alternative expressions of transformation matrices involving row vectors that is preferred by some authors eg.

40.1 Uses

Matrices allow arbitrary linear transformations to be represented in a consistent format, suitable for computation.[1] This also allows transformations to be concatenated easily (by multiplying their matrices). Linear transformations are not the only ones that can be represented by matrices. Some transformations that are non- linear on a n-dimensional Euclidean space Rn, can be represented as linear transformations on the n+1-dimensional space Rn+1. These include both affine transformations (such as translation) and projective transformations. For this reason, 4×4 transformation matrices are widely used in 3D computer graphics. These n+1-dimensional transformation matrices are called, depending on their application, affine transformation matrices, projective transformation matrices, or more generally non-linear transformation matrices. With respect to an n-dimensional matrix, an n+1-dimensional matrix can be described as an augmented matrix. In the physical sciences, an active transformation is one which actually changes the physical position of a system, and makes sense even in the absence of a coordinate system whereas a passive transformation is a change in the coordinate description of the physical system (change of basis). The distinction between active and passive transformations is important. By default, by transformation, mathematicians usually mean active transformations, while physicists could mean either. Put differently, a passive transformation refers to description of the same object as viewed from two different coor- dinate frames.

40.2 Finding the matrix of a transformation

If one has a linear transformation T (x) in functional form, it is easy to determine the transformation matrix A by transforming each of the vectors of the standard basis by T, then inserting the result into the columns of a matrix. In other words,

[ ] A = T (⃗e1) T (⃗e2) ··· T (⃗en) For example, the function T (x) = 5x is a linear transformation. Applying the above process (suppose that n = 2 in this case) reveals that

255 256 CHAPTER 40. TRANSFORMATION MATRIX

[ ] 5 0 T (⃗x) = 5⃗x = 5I⃗x = ⃗x 0 5

It must be noted that the matrix representation of vectors and operators depends on the chosen basis; a similar matrix will result from an alternate basis. Nevertheless, the method to find the components remains the same. T To elaborate, vector v can be represented in basis vectors, E = [⃗e1⃗e2 . . .⃗en] with coordinates [v]E = [v1v2 . . . vn] :

∑ ⃗v = v1⃗e1 + v2⃗e2 + ... + vn⃗en = vi⃗ei = E[v]E

Now, express the result of the transformation matrix A upon ⃗v , in the given basis:

∑ ∑ A(⃗v) = A( vi⃗ei) = viA(⃗ei) = [A(⃗e1)A(⃗e2) ...A(⃗en)][v]E =

   a1,1 a1,2 . . . a1,n v1    a2,1 a2,2 . . . a2,n v2  · = A [v]E = [⃗e1⃗e2 . . .⃗en] . . . .  .   . . .. .  .  an,1 an,2 . . . an,n vn

T The ai,j elements of matrix A are determined for a given basis E by applying A to∑ every ⃗ej = [00 ... (vj = 1) ... 0] , and observing the response vector A⃗ej = a1,j⃗e1 + a2,j⃗e2 + ... + an,j⃗en = ai,j⃗ei . This equation defines the [2] wanted elements, ai,j , of j-th column of the matrix A.

40.2.1 Eigenbasis and diagonal matrix

Main articles: Diagonal matrix and Eigenvalues and eigenvectors

Yet, there is a special basis for an operator in which the components form a diagonal matrix and, thus, multiplication complexity∑ reduces to n. Being diagonal means that all coefficients ai,j but ai,i are zeros leaving only one term in the sum ai,j⃗ei above. The surviving diagonal elements, ai,i , are known as eigenvalues and designated with λi in the [3] defining equation, which reduces to A⃗ei = λi⃗ei . The resulting equation is known as eigenvalue equation. The eigenvectors and eigenvalues are derived from it via the characteristic polynomial. With diagonalization, it is often possible to translate to and from eigenbases.

40.3 Examples in 2D computer graphics

Most common geometric transformations that keep the origin fixed are linear, including rotation, scaling, shearing, reflection, and orthogonal projection; if an affine transformation is not a pure translation it keeps some point fixed, and that point can be chosen as origin to make the transformation linear. In two dimensions, linear transformations can be represented using a 2×2 transformation matrix.

40.3.1 Rotation

For rotation by an angle θ clockwise about the origin the functional form is x′ = x cos θ+y sin θ and y′ = −x sin θ+ y cos θ . Written in matrix form, this becomes:[4]

[ ] [ ][ ] x′ cos θ sin θ x = y′ − sin θ cos θ y 40.4. EXAMPLES IN 3D COMPUTER GRAPHICS 257

Similarly, for a rotation counter-clockwise about the origin, the functional form is x′ = x cos θ − y sin θ and y′ = x sin θ + y cos θ and the matrix form is:

[ ] [ ][ ] x′ cos θ − sin θ x = y′ sin θ cos θ y

These formulae assume that the x axis points right and the y axis points up. In formats such as SVG where the y axis points down, these matrices must be swapped.

40.3.2 Shearing

For shear mapping (visually similar to slanting), there are two possibilities. A shear parallel to the x axis has x′ = x + ky and y′ = y . Written in matrix form, this becomes:

[ ] [ ][ ] x′ 1 k x = y′ 0 1 y

A shear parallel to the y axis has x′ = x and y′ = y + kx , which has matrix form:

[ ] [ ][ ] x′ 1 0 x = y′ k 1 y

40.3.3 Reflection

Main article: Householder transformation

To reflect a vector about a line that goes through the origin, let ⃗l=(lx,ly ) be a vector in the direction of the line:

[ ] 2 − 2 1 lx ly 2lxly A = 2 − 2 ∥⃗l∥2 2lxly ly lx

40.3.4 Orthogonal projection

To project a vector orthogonally onto a line that goes through the origin, let ⃗u = (ux,uy ) be a vector in the direction of the line. Then use the transformation matrix:

[ ] 2 1 ux uxuy A = 2 2 ∥⃗u∥ uxuy uy As with reflections, the orthogonal projection onto a line that does not pass through the origin is an affine, not linear, transformation. Parallel projections are also linear transformations and can be represented simply by a matrix. However, perspective projections are not, and to represent these with a matrix, homogeneous coordinates must be used.

40.4 Examples in 3D computer graphics

40.4.1 Rotation

The matrix to rotate an angle θ about the axis defined by unit vector (l,m,n) is[5] 258 CHAPTER 40. TRANSFORMATION MATRIX

  ll(1 − cos θ) + cos θ ml(1 − cos θ) − n sin θ nl(1 − cos θ) + m sin θ lm(1 − cos θ) + n sin θ mm(1 − cos θ) + cos θ nm(1 − cos θ) − l sin θ. ln(1 − cos θ) − m sin θ mn(1 − cos θ) + l sin θ nn(1 − cos θ) + cos θ

40.4.2 Reflection

Main article: Householder transformation

To reflect a point through a plane ax + by + cz = 0 (which goes through the origin), one can use A = I − 2NNT , where I is the 3x3 identity matrix and N is the three-dimensional unit vector for the vector normal of the plane. If the L2 norm of a, b, and c is unity, the transformation matrix can be expressed as:

  1 − 2a2 −2ab −2ac A =  −2ab 1 − 2b2 −2bc  −2ac −2bc 1 − 2c2

Note that these are particular cases of a Householder reflection in two and three dimensions. A reflection about a line or plane that does not go through the origin is not a linear transformation; it is an affine transformation.

40.5 Composing and inverting transformations

One of the main motivations for using matrices to represent linear transformations is that transformations can then be easily composed (combined) and inverted. Composition is accomplished by matrix multiplication. If A and B are the matrices of two linear transformations, then the effect of applying first A and then B to a vector x is given by:

B(A⃗x) = (BA)⃗x

(This is called the associative property.) In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. Note that the multiplication is done in the opposite order from the English sentence: the matrix of "A followed by B" is BA, not AB. A consequence of the ability to compose transformations by multiplying their matrices is that transformations can also be inverted by simply inverting their matrices. So, A−1 represents the transformation that “undoes” A.

40.6 Other kinds of transformations

40.6.1 Affine transformations

To represent affine transformations with matrices, we can use homogeneous coordinates. This means representing a 2-vector (x, y) as a 3-vector (x, y, 1), and similarly for higher dimensions. Using this system, translation can be ′ ′ expressed with matrix multiplication. The functional form x = x + tx; y = y + ty becomes:

     ′ x 1 0 tx x  ′    y = 0 1 ty y . 1 0 0 1 1

All ordinary linear transformations are included in the set of affine transformations, and can be described as a sim- plified form of affine transformations. Therefore, any linear transformation can also be represented by a general transformation matrix. The latter is obtained by expanding the corresponding linear transformation matrix by one 40.6. OTHER KINDS OF TRANSFORMATIONS 259

No change Translate Scale about origin 1 0 0 1 0 X W 0 0 0 1 0 0 1 Y 0 H 0 0 0 1 0 0 1 0 0 1 y y y (0,1) (0,H) 1 1 1 (0,0) (1,0) (X,Y) (W,0) 0 1x 0 1x 0 1x

Rotate about origin Shear in x direction Shear in y direction cos θ sin θ 0 1 A 0 1 0 0 -sin θ cos θ 0 0 1 0 B 1 0 0 0 1 0 0 1 0 0 1 y y y (sin θ, cos θ) (A,1) (0,1) 1 1 1 θ (1,0) (1,B) 0 1x 0 1x 0 1x (cos θ, -sin θ)

Reflect about origin Reflect about x-axis Reflect about y-axis -1 0 0 1 0 0 -1 0 0 0 -1 0 0 -1 0 0 1 0 0 0 1 0 0 1 0 0 1 y y y (0,1) 1 1 1 (-1,0) (1,0) 0 1x 0 1x (-1,0) 0 1x (0,-1) (0,-1)

Effect of applying various 2D affine transformation matrices on a unit square. Note that the reflection matrices are special cases of the scaling matrix.

row and column, filling the extra space with zeros except for the lower-right corner, which must be set to 1. For example, the clockwise rotation matrix from above becomes: 260 CHAPTER 40. TRANSFORMATION MATRIX

Affine transformations on the 2D plane can be performed in three dimensions. Translation is done by shearing along over the z axis, and rotation is performed around the z axis.

  cos θ sin θ 0 − sin θ cos θ 0 0 0 1

Using transformation matrices containing homogeneous coordinates, translations can be seamlessly intermixed with all other types of transformations. The reason is that the real plane is mapped to the w = 1 plane in real projective space, and so translation in real Euclidean space can be represented as a shear in real projective space. Although a translation is a non-linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates, it becomes, in a 3-D or 4-D projective space described by homogeneous coordinates, a simple linear transformation (a shear). More affine transformations can be obtained by composition of two or more affine transformations. For example, ′ ′ given a translation T' with vector (tx, ty) , a rotation R by an angle θ counter-clockwise, a scaling S with factors [6] (sx, sy) and a translation T of vector (tx, ty) , the result M of T'RST is: 40.7. SEE ALSO 261

  − − ′ sx cos θ sy sin θ txsx cos θ tysy sin θ + tx  ′  sx sin θ sy cos θ txsx sin θ + tysy cos θ + ty 0 0 1

When using affine transformations, the homogeneous component of a coordinate vector (normally called w) will never be altered. One can therefore safely assume that it is always 1 and ignore it. However, this is not true when using perspective projections.

40.6.2 Perspective projection

See also: Perspective projection

Another type of transformation, of importance in 3D computer graphics, is the perspective projection. Whereas parallel projections are used to project points onto the image plane along parallel lines, the perspective projection projects points onto the image plane along lines that emanate from a single point, called the center of projection. This means that an object has a smaller projection when it is far away from the center of projection and a larger projection when it is closer. The simplest perspective projection uses the origin as the center of projection, and z = 1 as the image plane. The functional form of this transformation is then x′ = x/z ; y′ = y/z . We can express this in homogeneous coordinates as:

     xc 1 0 0 0 x      yc  0 1 0 0y    =    zc 0 0 1 0 z wc 0 0 1 0 w

After carrying out the matrix multiplication, the homogeneous component wc will, in general, not be equal to 1. Therefore, to map back into the real plane we must perform the homogeneous divide or perspective divide by dividing each component by wc:

    ′ x xc  ′   y  1 yc   ′ =   z wc zc 1 wc

More complicated perspective projections can be composed by combining this one with rotations, scales, translations, and shears to move the image plane and center of projection wherever they are desired.

40.7 See also

• 3D projection

• Transformation (function)

40.8 References

[1] Gentle, James E. (2007). “Matrix Transformations and Factorizations”. Matrix Algebra: Theory, Computations, and Ap- plications in Statistics. Springer. ISBN 9780387708737.

[2] Nearing, James (2010). “Chapter 7.3 Examples of Operators” (PDF). Mathematical Tools for Physics. ISBN 048648212X. Retrieved January 1, 2012. 262 CHAPTER 40. TRANSFORMATION MATRIX

[3] Nearing, James (2010). “Chapter 7.9: Eigenvalues and Eigenvectors” (PDF). Mathematical Tools for Physics. ISBN 048648212X. Retrieved January 1, 2012.

[4] http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec03. pdf

[5] Szymanski, John E. (1989). Basic Mathematics for Electronic Engineers:Models and Applications. Taylor & Francis. p. 154. ISBN 0278000681.

[6] 2D transformation matrices baking -

40.9 External links

• The Matrix Page Practical examples in POV-Ray

• Reference page - Rotation of axes • Linear Transformation Calculator

• Transformation Applet - Generate matrices from 2D transformations and vice versa. • Coordinate transformation under rotation in 2D

• Excel Fun - Build 3D graphics from a spreadsheet - Interactive transformation matrices in a live spreadsheet Chapter 41

Translation (geometry)

A translation moves every point of a figure or a space by the same amount in a given direction.

In Euclidean geometry, a translation is a function that moves every point a constant distance in a specified direction. (Also in Euclidean geometry a transformation is a one to one correspondence between two sets of points or a mapping from one plane to another.[1]) A translation can be described as a rigid motion: other rigid motions include rotations

263 264 CHAPTER 41. TRANSLATION (GEOMETRY)

B’C’

A’

M/2

M

CB

A

A reflection against an axis followed by a reflection against a second axis parallel to the first one results in a total motion which is a translation.

and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system.A translation operator is an operator Tδ such that Tδf(v) = f(v + δ). If v is a fixed vector, then the translation Tᵥ will work as Tᵥ(p) = p + v. If T is a translation, then the image of a subset A under the function T is the translate of A by T. The translate of A by Tᵥ is often written A + v. In a Euclidean space, any translation is an isometry. The set of all translations forms the translation group T, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E(n ). The quotient group of E(n ) by T is isomorphic to the orthogonal group O(n ):

E(n ) /T ≅ O(n ). 41.1. MATRIX REPRESENTATION 265

41.1 Matrix representation

A translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point. Nevertheless, there is a common workaround using homogeneous coordinates to represent a translation of a vector space with matrix multiplication: Write the 3-dimensional vector w = (wx, wy, wz) using 4 homogeneous coordinates as w = (wx, wy, wz, 1).[2] To translate an object by a vector v, each homogeneous vector p (written in homogeneous coordinates) can be mul- tiplied by this translation matrix:

  1 0 0 vx   0 1 0 vy Tv =   0 0 1 vz 0 0 0 1 As shown below, the multiplication will give the expected result:

     1 0 0 vx px px + vx      0 1 0 vypy py + vy  Tvp =    =   = p + v 0 0 1 vz pz pz + vz 0 0 0 1 1 1 The inverse of a translation matrix can be obtained by reversing the direction of the vector:

−1 Tv = T−v. Similarly, the product of translation matrices is given by adding the vectors:

TuTv = Tu+v. Because addition of vectors is commutative, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).

41.2 Translations in physics

In physics, translation (Translational motion) is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:[3]

If a body is moved from one position to another, and if the lines joining the initial and final points of each of the points of the body are a set of parallel straight lines of length ℓ, so that the orientation of the body in space is unaltered, the displacement is called a translation parallel to the direction of the lines, through a distance ℓ. — E. T. Whittaker: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, p. 1

A translation is the operation changing the positions of all points (x, y, z) of an object according to the formula

(x, y, z) → (x + ∆x, y + ∆y, z + ∆z) where (∆x, ∆y, ∆z) is the same vector for each point of the object. The translation vector (∆x, ∆y, ∆z) common to all points of the object describes a particular type of displacement of the object, usually called a linear displacement to distinguish it from displacements involving rotation, called angular displacements. When considering spacetime, a change of time coordinate is considered to be a translation. For example, the Galilean group and the Poincaré group include translations with respect to time. 266 CHAPTER 41. TRANSLATION (GEOMETRY)

41.3 See also

• Advection

• Rotation matrix • Scaling (geometry)

• Transformation matrix

• Translation of axes • Translational symmetry

• Vertical translation

41.4 External links

• Translation Transform at cut-the-knot

• Geometric Translation (Interactive Animation) at Math Is Fun • Understanding 2D Translation and Understanding 3D Translation by Roger Germundsson, The Wolfram Demon- strations Project.

41.5 References

[1] Osgood, William F. & Graustein, William C. (1921). Plane and solid analytic geometry. The Macmillan Company. p. 330.

[2] Richard Paul, 1981, Robot manipulators: mathematics, programming, and control : the computer control of robot manip- ulators, MIT Press, Cambridge, MA

[3] Edmund Taylor Whittaker (1988). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Reprint of fourth edition of 1936 with foreword by William McCrea ed.). Cambridge University Press. p. 1. ISBN 0-521-35883-3. Chapter 42

Vector space

This article is about linear (vector) spaces. For the structure in incidence geometry, see Linear space (geometry). A vector space (also called a linear space) is a collection of objects called vectors, which may be added together

v v+w w

v v+2w 2w

Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.

and multiplied (“scaled”) by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below. Euclidean vectors are an example of a vector space. They represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow- like objects as they appear in the mentioned examples: vectors are regarded as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows. Vector spaces are the subject of linear algebra and are well understood from this point of view since vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. A vector space may be endowed with additional structure, such as a norm or inner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide whether a sequence of vectors converges to a given

267 268 CHAPTER 42. VECTOR SPACE

vector. This is accomplished by considering vector spaces with additional structure, mostly spaces endowed with a suitable topology, thus allowing the consideration of proximity and continuity issues. These topological vector spaces, in particular Banach spaces and Hilbert spaces, have a richer theory. Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century’s analytic geometry, matrices, systems of linear equations, and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in 1888, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs. Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear- algebraic notion to deal with systems of linear equations; offer a framework for Fourier expansion, which is employed in image compression routines; or provide an environment that can be used for solution techniques for partial differ- ential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra.

42.1 Introduction and definition

The concept of vector space will first be explained by describing two particular examples:

42.1.1 First example: arrows in the plane

The first example of a vector space consists of arrows in a fixed plane, starting at one fixed point. This is used in physics to describe forces or velocities. Given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w. In the special case of two arrows on the same line, their sum is the arrow on this line whose length is the sum or the difference of the lengths, depending on whether the arrows have the same direction. Another operation that can be done with arrows is scaling: given any positive real number a, the arrow that has the same direction as v, but is dilated or shrunk by multiplying its length by a, is called multiplication of v by a. It is denoted av. When a is negative, av is defined as the arrow pointing in the opposite direction, instead. The following shows a few examples: if a = 2, the resulting vector aw has the same direction as w, but is stretched to the double length of w (right image below). Equivalently 2w is the sum w + w. Moreover, (−1)v = −v has the opposite direction and the same length as v (blue vector pointing down in the right image).

42.1.2 Second example: ordered pairs of numbers

A second key example of a vector space is provided by pairs of real numbers x and y. (The order of the components x and y is significant, so such a pair is also called an ordered pair.) Such a pair is written as (x, y). The sum of two such pairs and multiplication of a pair with a number is defined as follows:

(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)

and

a (x, y) = (ax, ay).

The first example above reduces to this one if the arrows are represented by the pair of Cartesian coordinates of their end points.

42.1.3 Definition

A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below. Elements of V are commonly called vectors. Elements of F are commonly called scalars. The first operation, called 42.2. HISTORY 269

vector addition or simply addition, takes any two vectors v and w and assigns to them a third vector which is commonly written as v + w, and called the sum of these two vectors. The second operation, called scalar multiplication takes any scalar a and any vector v and gives another vector av. In this article, vectors are distinguished from scalars by boldface.[nb 1] In the two examples above, the field is the field of the real numbers and the set of the vectors consists of the planar arrows with fixed starting point and of pairs of real numbers, respectively. To qualify as a vector space, the set V and the operations of addition and multiplication must adhere to a number of requirements called axioms.[1] In the list below, let u, v and w be arbitrary vectors in V, and a and b scalars in F. These axioms generalize properties of the vectors introduced in the above examples. Indeed, the result of addition of two ordered pairs (as in the second example above) does not depend on the order of the summands:

(xᵥ, yᵥ) + (x, y) = (x, y) + (xᵥ, yᵥ).

Likewise, in the geometric example of vectors as arrows, v + w = w + v since the parallelogram defining the sum of the vectors is independent of the order of the vectors. All other axioms can be checked in a similar manner in both examples. Thus, by disregarding the concrete nature of the particular type of vectors, the definition incorporates these two and many more examples in one notion of vector space. Subtraction of two vectors and division by a (non-zero) scalar can be defined as

v − w = v + (−w), v/a = (1/a)v.

When the scalar field F is the real numbers R, the vector space is called a real vector space. When the scalar field is the complex numbers, it is called a complex vector space. These two cases are the ones used most often in engineering. The general definition of a vector space allows scalars to be elements of any fixed field F. The notion is then known as an F-vector spaces or a vector space over F. A field is, essentially, a set of numbers possessing addition, subtraction, multiplication and division operations.[nb 3] For example, rational numbers also form a field. In contrast to the intuition stemming from vectors in the plane and higher-dimensional cases, there is, in general vector spaces, no notion of nearness, angles or distances. To deal with such matters, particular types of vector spaces are introduced; see below.

42.1.4 Alternative formulations and elementary consequences

The requirement that vector addition and scalar multiplication be binary operations includes (by definition of binary operations) a property called closure: that u + v and av are in V for all a in F, and u, v in V. Some older sources mention these properties as separate axioms.[2] In the parlance of abstract algebra, the first four axioms can be subsumed by requiring the set of vectors to be an abelian group under addition. The remaining axioms give this group an F-module structure. In other words, there is a ring homomorphism f from the field F into the endomorphism ring of the group of vectors. Then scalar multiplication av is defined as (f(a))(v).[3] There are a number of direct consequences of the vector space axioms. Some of them derive from elementary group theory, applied to the additive group of vectors: for example the zero vector 0 of V and the additive inverse −v of any vector v are unique. Other properties follow from the distributive law, for example av equals 0 if and only if a equals 0 or v equals 0.

42.2 History

Vector spaces stem from affine geometry via the introduction of coordinates in the plane or three-dimensional space. Around 1636, Descartes and Fermat founded analytic geometry by equating solutions to an equation of two variables with points on a plane curve.[4] In 1884, to achieve geometric solutions without using coordinates, Bolzano introduced certain operations on points, lines and planes, which are predecessors of vectors.[5] His work was then used in the conception of barycentric coordinates by Möbius in 1827.[6] The definition of vectors was founded on Bellavitis' notion 270 CHAPTER 42. VECTOR SPACE

of the bipoint, an oriented segment of which one end is the origin and the other a target, then further elaborated with the presentation of complex numbers by Argand and Hamilton and the introduction of quaternions and biquaternions by the latter.[7] They are elements in R2, R4, and R8; their treatment as linear combinations can be traced back to Laguerre in 1867, who also defined systems of linear equations. In 1857, Cayley introduced matrix notation, which allows for a harmonization and simplification of linear maps. Around the same time, Grassmann studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations.[8] In his work, the concepts of linear independence and dimension, as well as scalar products, are present. In fact, Grassmann’s 1844 work exceeds the framework of vector spaces, since his consideration of multiplication led him to what are today called algebras. Peano was the first to give the modern definition of vector spaces and linear maps in 1888.[9] An important development of vector spaces is due to the construction of function spaces by Lebesgue. This was later formalized by Banach and Hilbert, around 1920.[10] At that time, algebra and the new field of functional analysis began to interact, notably with key concepts such as spaces of p-integrable functions and Hilbert spaces.[11] Vector spaces, including infinite-dimensional ones, then became a firmly established notion, and many mathematical branches started making use of this concept.

42.3 Examples

Main article: Examples of vector spaces

42.3.1 Coordinate spaces

Main article: Coordinate space

The most simple example of a vector space over a field F is the field itself, equipped with its standard addition and multiplication. More generally, a vector space can be composed of n-tuples (sequences of length n) of elements of F, such as

[12] (a1, a2, ..., an), where each ai is an element of F.

A vector space composed of all the n-tuples of a field F is known as a coordinate space, usually denoted Fn. The case n = 1 is the above-mentioned simplest example, in which the field F is also regarded as a vector space over itself. The case F = R and n = 2 was discussed in the introduction above.

42.3.2 Complex numbers and other field extensions

The set of complex numbers C, i.e., numbers that can be written in the form x + iy for real numbers x and y where i is the imaginary unit, form a vector space over the reals with the usual addition and multiplication: (x + iy) + (a + ib) = (x + a) + i(y + b) and c ⋅ (x + iy) = (c ⋅ x) + i(c ⋅ y) for real numbers x, y, a, b and c. The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic. In fact, the example of complex numbers is essentially the same (i.e., it is isomorphic) to the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number x + i y as representing the ordered pair (x, y) in the complex plane then we see that the rules for sum and scalar product correspond exactly to those in the earlier example. More generally, field extensions provide another class of examples of vector spaces, particularly in algebra and algebraic number theory: a field F containing a smaller field E is an E-vector space, by the given multiplication and addition√ operations of F.[13] For example, the complex numbers are a vector space over R, and the field extension Q(i 5) is a vector space over Q. 42.4. BASIS AND DIMENSION 271

42.3.3 Function spaces

Functions from any fixed set Ω to a field F also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions f and g is the function (f + g) given by

(f + g)(w) = f(w) + g(w), and similarly for multiplication. Such function spaces occur in many geometric situations, when Ω is the real line or an interval, or other subsets of R. Many notions in topology and analysis, such as continuity, integrability or differentiability are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property.[14] Therefore, the set of such functions are vector spaces. They are studied in greater detail using the methods of functional analysis, see below. Algebraic constraints also yield vector spaces: the vector space F[x] is given by polynomial functions:

n−1 n [15] f(x) = r0 + r1x + ... + rn₋₁x + rnx , where the coefficients r0, ..., rn are in F.

42.3.4 Linear equations

Main articles: Linear equation, Linear differential equation and Systems of linear equations

Systems of homogeneous linear equations are closely tied to vector spaces.[16] For example, the solutions of

are given by triples with arbitrary a, b = a/2, and c = −5a/2. They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too. Matrices can be used to condense multiple linear equations as above into one vector equation, namely

Ax = 0, [ ] 1 3 1 where A = is the matrix containing the coefficients of the given equations, x is the vector (a, b, c), Ax 4 2 2 denotes the matrix product, and 0 = (0, 0) is the zero vector. In a similar vein, the solutions of homogeneous linear differential equations form vector spaces. For example,

f′′(x) + 2f′(x) + f(x) = 0 yields f(x) = a e−x + bx e−x, where a and b are arbitrary constants, and ex is the natural exponential function.

42.4 Basis and dimension

Main articles: Basis and Dimension Bases allow to represent vectors by a sequence of scalars called coordinates or components. A basis is a (finite or infinite) set B = {bi}i ∈ I of vectors bi, for convenience often indexed by some index set I, that spans the whole space and is linearly independent. “Spanning the whole space” means that any vector v can be expressed as a finite sum (called a linear combination) of the basis elements:

where the ak are scalars, called the coordinates (or the components) of the vector v with respect to the basis B, and bik (k = 1, ..., n) elements of B. Linear independence means that the coordinates ak are uniquely determined for any vector in the vector space. n For example, the coordinate vectors e1 = (1, 0, ..., 0), e2 = (0, 1, 0, ..., 0), to en = (0, 0, ..., 0, 1), form a basis of F , called the standard basis, since any vector (x1, x2, ..., xn) can be uniquely expressed as a linear combination of these vectors: 272 CHAPTER 42. VECTOR SPACE

2 2 A vector v in R (blue) expressed in terms of different bases: using the standard basis of R v = xe1 + ye2 (black), and using a different, non-orthogonal basis: v = f1 + f2 (red).

(x1, x2, ..., xn) = x1(1, 0, ..., 0) + x2(0, 1, 0, ..., 0) + ... + xn(0, ..., 0, 1) = x1e1 + x2e2 + ... + xnen.

The corresponding coordinates x1, x2, ..., xn are just the Cartesian coordinates of the vector. Every vector space has a basis. This follows from Zorn’s lemma, an equivalent formulation of the Axiom of Choice.[17] Given the other axioms of Zermelo–Fraenkel set theory, the existence of bases is equivalent to the axiom of choice.[18] The ultrafilter lemma, which is weaker than the axiom of choice, implies that all bases of a given vector space have the same number of elements, or cardinality (cf. Dimension theorem for vector spaces).[19] It is called the dimension of the vector space, denoted dim V. If the space is spanned by finitely many vectors, the above statements can be proven without such fundamental input from set theory.[20] The dimension of the coordinate space Fn is n, by the basis exhibited above. The dimension of the polynomial ring F[x] introduced above is countably infinite, a basis is given by 1, x, x2, ... A fortiori, the dimension of more general function spaces, such as the space of functions on some (bounded or unbounded) interval, is infinite.[nb 4] Under suitable regularity assumptions on the coefficients involved, the dimension of the solution space of a homogeneous ordinary differential equation equals the degree of the equation.[21] For example, the solution space for the above equation is generated by e−x and xe−x. These two functions are linearly independent over R, so the dimension of this space is two, as is the degree of the equation. 42.5. LINEAR MAPS AND MATRICES 273

A field extension over the rationals Q can be thought of as a vector space over Q (by defining vector addition as field addition, defining scalar multiplication as field multiplication by elements of Q, and otherwise ignoring the field multiplication). The dimension (or degree) of the field extension Q(α) over Q depends on α. If α satisfies some polynomial equation

n n−1 qnα + qn₋₁α + ... + q0 = 0, with rational coefficients qn, ..., q0.

("α is algebraic"), the dimension is finite. More precisely, it equals the degree of the minimal polynomial having α as a root.[22] For example, the complex numbers C are a two-dimensional real vector space, generated by 1 and the imaginary unit i. The latter satisfies i2 + 1 = 0, an equation of degree two. Thus, C is a two-dimensional R-vector space (and, as any field, one-dimensional as a vector space over itself, C). If α is not algebraic, the dimension of Q(α) over Q is infinite. For instance, for α = π there is no such equation, in other words π is transcendental.[23]

42.5 Linear maps and matrices

Main article: Linear map

The relation of two vector spaces can be expressed by linear map or linear transformation. They are functions that reflect the vector space structure—i.e., they preserve sums and scalar multiplication:

f(x + y) = f(x) + f(y) and f(a · x) = a · f(x) for all x and y in V, all a in F.[24]

An isomorphism is a linear map f : V → W such that there exists an inverse map g : W → V, which is a map such that the two possible compositions f ∘ g : W → W and g ∘ f : V → V are identity maps. Equivalently, f is both one-to-one (injective) and onto (surjective).[25] If there exists an isomorphism between V and W, the two spaces are said to be isomorphic; they are then essentially identical as vector spaces, since all identities holding in V are, via f, transported to similar ones in W, and vice versa via g. For example, the “arrows in the plane” and “ordered pairs of numbers” vector spaces in the introduction are isomor- phic: a planar arrow v departing at the origin of some (fixed) coordinate system can be expressed as an ordered pair by considering the x- and y-component of the arrow, as shown in the image at the right. Conversely, given a pair (x, y), the arrow going by x to the right (or to the left, if x is negative), and y up (down, if y is negative) turns back the arrow v. Linear maps V → W between two vector spaces form a vector space HomF(V, W), also denoted L(V, W).[26] The space of linear maps from V to F is called the dual vector space, denoted V∗.[27] Via the injective natural map V → V∗∗, any vector space can be embedded into its bidual; the map is an isomorphism if and only if the space is finite-dimensional.[28] Once a basis of V is chosen, linear maps f : V → W are completely determined by specifying the images of the basis vectors, because any element of V is expressed uniquely as a linear combination of them.[29] If dim V = dim W, a 1-to-1 correspondence between fixed bases of V and W gives rise to a linear map that maps any basis element of V to the corresponding basis element of W. It is an isomorphism, by its very definition.[30] Therefore, two vector spaces are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space is completely classified (up to isomorphism) by its dimension, a single number. In particular, any n-dimensional F-vector space V is isomorphic to Fn. There is, however, no “canonical” or preferred isomorphism; actually an isomorphism φ : Fn → V is equivalent to the choice of a basis of V, by mapping the standard basis of Fn to V, via φ. The freedom of choosing a convenient basis is particularly useful in the infinite-dimensional context, see below.

42.5.1 Matrices

Main articles: Matrix and Determinant Matrices are a useful notion to encode linear maps.[31] They are written as a rectangular array of scalars as in the image at the right. Any m-by-n matrix A gives rise to a linear map from Fn to Fm, by the following

(∑ ∑ ∑ ) ∑ ··· 7→ n n ··· n x = (x1, x2, , xn) j=1 a1jxj, j=1 a2jxj, , j=1 amjxj , where denotes summation, 274 CHAPTER 42. VECTOR SPACE

Describing an arrow vector v by its coordinates x and y yields an isomorphism of vector spaces.

or, using the matrix multiplication of the matrix A with the coordinate vector x:

x ↦ Ax.

Moreover, after choosing bases of V and W, any linear map f : V → W is uniquely represented by a matrix via this assignment.[32] The determinant det (A) of a square matrix A is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero.[33] The linear transformation of Rn corresponding to a real n-by-n matrix is orientation preserving if and only if its determinant is positive.

42.5.2 Eigenvalues and eigenvectors

Main article: Eigenvalues and eigenvectors

Endomorphisms, linear maps f : V → V, are particularly important since in this case vectors v can be compared with their image under f, f(v). Any nonzero vector v satisfying λv = f(v), where λ is a scalar, is called an eigenvector of f with eigenvalue λ.[nb 5][34] Equivalently, v is an element of the kernel of the difference f − λ · Id (where Id is the identity map V → V). If V is finite-dimensional, this can be rephrased using determinants: f having eigenvalue λ is equivalent to

det(f − λ · Id) = 0.

By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in λ, called the characteristic polynomial of f.[35] If the field F is large enough to contain a zero of this 42.6. BASIC CONSTRUCTIONS 275

ai,j n columns j changes m rows . . a1,1 a1,2 a1,3 . i c . . a2,1 . h a2,2 a2,3 a n . . a3,1 a3,2 a3,3 . g e . . . . s ......

A typical matrix

polynomial (which automatically happens for F algebraically closed, such as F = C) any linear map has at least one eigenvector. The vector space V may or may not possess an eigenbasis, a basis consisting of eigenvectors. This phenomenon is governed by the Jordan canonical form of the map.[nb 6] The set of all eigenvectors corresponding to a particular eigenvalue of f forms a vector space known as the eigenspace corresponding to the eigenvalue (and f) in question. To achieve the spectral theorem, the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see below.

42.6 Basic constructions

In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones. In addition to the definitions given below, they are also characterized by universal properties, which determine an object X by specifying the linear maps from X to any other vector space.

42.6.1 Subspaces and quotient spaces

Main articles: Linear subspace and Quotient vector space A nonempty subset W of a vector space V that is closed under addition and scalar multiplication (and therefore contains the 0-vector of V) is called a subspace of V.[36] Subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set S of vectors is called its span, and it is the smallest subspace of V containing the set S. Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of S.[37] The counterpart to subspaces are quotient vector spaces.[38] Given any subspace W ⊂ V, the quotient space V/W ("V modulo W") is defined as follows: as a set, it consists of v + W = {v + w : w ∈ W}, where v is an arbitrary vector in 276 CHAPTER 42. VECTOR SPACE

The volume of this parallelepiped is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors r1, r2, and r3.

V. The sum of two such elements v1 + W and v2 + W is (v1 + v2) + W, and scalar multiplication is given by a ·(v + W) = (a · v) + W. The key point in this definition is that v1 + W = v2 + W if and only if the difference of v1 and v2 lies in W.[nb 7] This way, the quotient space “forgets” information that is contained in the subspace W. The kernel ker(f) of a linear map f : V → W consists of vectors v that are mapped to 0 in W.[39] Both kernel and image im(f) = {f(v): v ∈ V} are subspaces of V and W, respectively.[40] The existence of kernels and images is part of the statement that the category of vector spaces (over a fixed field F) is an abelian category, i.e. a corpus of mathematical objects and structure-preserving maps between them (a category) that behaves much like the category of abelian groups.[41] Because of this, many statements such as the first isomorphism theorem (also called rank–nullity theorem in matrix-related terms)

V / ker(f) ≡ im(f). and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corre- sponding statements for groups. An important example is the kernel of a linear map x ↦ Ax for some fixed matrix A, as above. The kernel of this map is the subspace of vectors x such that Ax = 0, which is precisely the set of solutions to the system of homogeneous linear equations belonging to A. This concept also extends to linear differential equations

df d2f ··· dnf a0f + a1 dx + a2 dx2 + + an dxn = 0 , where the coefficients ai are functions in x, too.

In the corresponding map 42.6. BASIC CONSTRUCTIONS 277

A line passing through the origin (blue, thick) in R3 is a linear subspace. It is the intersection of two planes (green and yellow).

∑n dif f 7→ D(f) = a i dxi i=0 the derivatives of the function f appear linearly (as opposed to f′′(x)2, for example). Since differentiation is a linear procedure (i.e., (f + g)′ = f′ + g ′ and (c·f)′ = c·f′ for a constant c) this assignment is linear, called a linear differential operator. In particular, the solutions to the differential equation D(f) = 0 form a vector space (over R or C).

42.6.2 Direct product and direct sum

Main articles: Direct product and Direct sum of modules

The direct product of vector spaces and the direct sum of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space. ∏ The direct product i∈I Vi of a family of vector spaces Vi consists of the set of all tuples (vi)i ∈ I, which specify for [42] each index i in some index set I an element vi of Vi. Addition and scalar multiplication is performed⨿ componen- ⊕ twise. A variant of this construction is the direct sum i∈I Vi (also called coproduct and denoted i∈I Vi ), where only tuples with finitely many nonzero vectors are allowed. If the index set I is finite, the two constructions agree, but in general they are different.

42.6.3 Tensor product

Main article: Tensor product of vector spaces

The tensor product V ⊗FW, or simply V ⊗ W, of two vector spaces V and W is one of the central notions of multilinear algebra which deals with extending notions such as linear maps to several variables. A map g : V × W → 278 CHAPTER 42. VECTOR SPACE

X is called bilinear if g is linear in both variables v and w. That is to say, for fixed w the map v ↦ g(v, w) is linear in the sense above and likewise for fixed v. The tensor product is a particular vector space that is a universal recipient of bilinear maps g, as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called tensors

v1 ⊗ w1 + v2 ⊗ w2 + ... + vn ⊗ wn,

subject to the rules

a ·(v ⊗ w) = (a · v) ⊗ w = v ⊗ (a · w), where a is a scalar,

(v1 + v2) ⊗ w = v1 ⊗ w + v2 ⊗ w, and [43] v ⊗ (w1 + w2) = v ⊗ w1 + v ⊗ w2.

Commutative diagram depicting the universal property of the tensor product.

These rules ensure that the map f from the V × W to V ⊗ W that maps a tuple (v, w) to v ⊗ w is bilinear. The universality states that given any vector space X and any bilinear map g : V × W → X, there exists a unique map u, shown in the diagram with a dotted arrow, whose composition with f equals g: u(v ⊗ w) = g(v, w).[44] This is called the universal property of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object.

42.7 Vector spaces with additional structure

From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space is characterized, up to isomorphism, by its dimension. However, vector spaces per se do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions converges to another function. Likewise, linear algebra is not adapted to deal with infinite series, since the addition operation allows only finitely many terms to be added. Therefore, the needs of functional analysis require considering additional structures. A vector space may be given a partial order ≤, under which some vectors can be compared.[45] For example, n- dimensional real space Rn can be ordered by comparing its vectors componentwise. Ordered vector spaces, for example Riesz spaces, are fundamental to Lebesgue integration, which relies on the ability to express a function as a difference of two positive functions 42.7. VECTOR SPACES WITH ADDITIONAL STRUCTURE 279

f = f+ − f−,

where f+ denotes the positive part of f and f− the negative part.[46]

42.7.1 Normed vector spaces and inner product spaces

Main articles: Normed vector space and Inner product space

“Measuring” vectors is done by specifying a norm, a datum which measures lengths of vectors, or by an inner product, which measures angles between vectors. Norms and inner products are denoted |v| and ⟨v, w⟩ , respectively. The √datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm |v| := ⟨v, v⟩ . Vector spaces endowed with such data are known as normed vector spaces and inner product spaces, respectively.[47] Coordinate space Fn can be equipped with the standard dot product:

⟨x, y⟩ = x · y = x1y1 + ··· + xnyn.

In R2, this reflects the common notion of the angle between two vectors x and y, by the law of cosines:

x · y = cos (∠(x, y)) · |x| · |y|.

Because of this, two vectors satisfying ⟨x, y⟩ = 0 are called orthogonal. An important variant of the standard dot product is used in Minkowski space: R4 endowed with the Lorentz product

[48] ⟨x|y⟩ = x1y1 + x2y2 + x3y3 − x4y4.

In contrast to the standard dot product, it is not positive definite: ⟨x|x⟩ also takes negative values, for example for x = (0, 0, 0, 1) . Singling out the fourth coordinate—corresponding to time, as opposed to three space-dimensions— makes it useful for the mathematical treatment of special relativity.

42.7.2 Topological vector spaces

Main article: Topological vector space

Convergence questions are treated by considering vector spaces V carrying a compatible topology, a structure that allows one to talk about elements being close to each other.[49][50] Compatible here means that addition and scalar multiplication have to be continuous maps. Roughly, if x and y in V, and a in F vary by a bounded amount, then so do x + y and ax.[nb 8] To make sense of specifying the amount a scalar changes, the field F also has to carry a topology in this context; a common choice are the reals or the complex numbers. In such topological vector spaces one can consider series of vectors. The infinite sum

∑∞ fi i=0 denotes the limit of the corresponding finite partial sums of the sequence (fi)i∈N of elements of V. For example, the fi could be (real or complex) functions belonging to some function space V, in which case the series is a function series. The mode of convergence of the series depends on the topology imposed on the function space. In such cases, pointwise convergence and uniform convergence are two prominent examples. A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any Cauchy sequence has a limit; such a vector space is called complete. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval [0,1], equipped with 280 CHAPTER 42. VECTOR SPACE

2 1

∞

2 Unit “spheres” in R consist of plane vectors of norm√ 1. Depicted are the unit spheres in different p-norms, for p = 1, 2, and ∞. The bigger diamond depicts points of 1-norm equal to 2 . the topology of uniform convergence is not complete because any continuous function on [0,1] can be uniformly approximated by a sequence of polynomials, by the Weierstrass approximation theorem.[51] In contrast, the space of all continuous functions on [0,1] with the same topology is complete.[52] A norm gives rise to a topology by defining that a sequence of vectors vn converges to v if and only if

limn→∞|vn − v| = 0. Banach and Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study—a key piece of functional analysis—focusses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence.[53] The image at the right shows the equivalence of the 1-norm and ∞-norm on R2: as the unit “balls” enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data. From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called functionals) V → W, maps between topological vector 42.7. VECTOR SPACES WITH ADDITIONAL STRUCTURE 281

spaces are required to be continuous.[54] In particular, the (topological) dual space V∗ consists of continuous function- als V → R (or to C). The fundamental Hahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.[55]

Banach spaces

Main article: Banach space

Banach spaces, introduced by Stefan Banach, are complete normed vector spaces.[56] A first example is the vector p space ℓ consisting of infinite vectors with real entries x = (x1, x2, ...) whose p-norm (1 ≤ p ≤ ∞) given by ∑ | | | |p 1/p | | | | x p := ( i xi ) for p < ∞ and x ∞ := supi xi is finite. The topologies on the infinite-dimensional space ℓ p are inequivalent for different p. E.g. the sequence of vectors xn = (2−n, 2−n, ..., 2−n, 0, 0, ...), i.e. the first 2n components are 2−n, the following ones are 0, converges to the zero vector for p = ∞, but does not for p = 1:

∑ n | | −n −n → | | 2 −n n · −n xn ∞ = sup(2 , 0) = 2 0 , but xn 1 = i=1 2 = 2 2 = 1.

More generally than sequences of real numbers, functions f: Ω → R are endowed with a norm that replaces the above sum by the Lebesgue integral

(∫ )1/p p |f|p := |f(x)| dx . Ω The space of integrable functions on a given domain Ω (for example an interval) satisfying |f|p < ∞, and equipped with this norm are called Lebesgue spaces, denoted Lp(Ω).[nb 9] These spaces are complete.[57] (If one uses the Riemann integral instead, the space is not complete, which may be seen as a justification for Lebesgue’s integration theory.[nb 10]) Concretely this means that for any sequence of Lebesgue-integrable functions f1, f2, ... with |fn|p < ∞, satisfying the condition

∫ p lim |fk(x) − fn(x)| dx = 0 →∞ k, n Ω there exists a function f(x) belonging to the vector space Lp(Ω) such that

∫ p lim |f(x) − fk(x)| dx = 0. →∞ k Ω Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.[58]

Hilbert spaces

Main article: Hilbert space Complete inner product spaces are known as Hilbert spaces, in honor of David Hilbert.[59] The Hilbert space L2(Ω), with inner product given by

∫ ⟨f , g⟩ = f(x)g(x) dx, Ω where g(x) denotes the complex conjugate of g(x),[60][nb 11] is a key case. By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions fn with desirable properties that approximates a given limit function, is equally crucial. Early analysis, in the guise 282 CHAPTER 42. VECTOR SPACE

The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).

of the Taylor approximation, established an approximation of differentiable functions f by polynomials.[61] By the Stone–Weierstrass theorem, every continuous function on [a, b] can be approximated as closely as desired by a polynomial.[62] A similar approximation technique by trigonometric functions is commonly called Fourier expansion, and is much applied in engineering, see below. More generally, and more conceptually, the theorem yields a simple description of what “basic functions”, or, in abstract Hilbert spaces, what basic vectors suffice to generate a Hilbert space H, in the sense that the closure of their span (i.e., finite linear combinations and limits of those) is the whole space. Such a set of functions is called a basis of H, its cardinality is known as the Hilbert space dimension.[nb 12] Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but together with the Gram-Schmidt process, it enables one to construct a basis of orthogonal vectors.[63] Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional Euclidean space. The solutions to various differential equations can be interpreted in terms of Hilbert spaces. For example, a great many fields in physics and engineering lead to such equations and frequently solutions with particular physical prop- erties are used as basis functions, often orthogonal.[64] As an example from physics, the time-dependent Schrödinger equation in quantum mechanics describes the change of physical properties in time by means of a partial differen- tial equation, whose solutions are called wavefunctions.[65] Definite values for physical properties such as energy, or momentum, correspond to eigenvalues of a certain (linear) differential operator and the associated wavefunctions are called eigenstates. The spectral theorem decomposes a linear compact operator acting on functions in terms of these eigenfunctions and their eigenvalues.[66]

42.7.3 Algebras over fields

Main articles: Algebra over a field and Lie algebra General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional bilinear operator defining the multiplication of two vectors is an algebra over a field.[67] Many algebras stem from functions on some geometrical object: since functions with values in a given field can be multiplied pointwise, these entities form algebras. The Stone–Weierstrass theorem mentioned above, for example, relies on Banach algebras which are both Banach spaces and algebras. Commutative algebra makes great use of rings of polynomials in one or several variables, introduced above. Their multiplication is both commutative and associative. These rings and their quotients form the basis of algebraic ge- ometry, because they are rings of functions of algebraic geometric objects.[68] Another crucial example are Lie algebras, which are neither commutative nor associative, but the failure to be so is limited by the constraints ([x, y] denotes the product of x and y):

• [x, y] = −[y, x](anticommutativity), and • [x,[y, z]] + [y,[z, x]] + [z,[x, y]] = 0 (Jacobi identity).[69]

Examples include the vector space of n-by-n matrices, with [x, y] = xy − yx, the commutator of two matrices, and R3, endowed with the cross product. The tensor algebra T(V) is a formal way of adding products to any vector space V to obtain an algebra.[70] As a vector space, it is spanned by symbols, called simple tensors

v1 ⊗ v2 ⊗ ... ⊗ vn, where the degree n varies. 42.8. APPLICATIONS 283

A hyperbola, given by the equation x y = 1. The coordinate ring of functions on this hyperbola is given by R[x, y]/(x · y − 1), an infinite-dimensional vector space over R.

The multiplication is given by concatenating such symbols, imposing the distributive law under addition, and requiring that scalar multiplication commute with the tensor product ⊗, much the same way as with the tensor product of two vector spaces introduced above. In general, there are no relations between v1 ⊗ v2 and v2 ⊗ v1. Forcing two such [71] elements to be equal leads to the symmetric algebra, whereas forcing v1 ⊗ v2 = − v2 ⊗ v1 yields the exterior algebra. When a field, F is explicitly stated, a common term used is F-algebra.

42.8 Applications

Vector spaces have manifold applications as they occur in many circumstances, namely wherever functions with val- ues in some field are involved. They provide a framework to deal with analytical and geometrical problems, or are used in the Fourier transform. This list is not exhaustive: many more applications exist, for example in optimization. The minimax theorem of game theory stating the existence of a unique payoff when all players play optimally can be formulated and proven using vector spaces methods.[72] Representation theory fruitfully transfers the good under- standing of linear algebra and vector spaces to other mathematical domains such as group theory.[73] 284 CHAPTER 42. VECTOR SPACE

42.8.1 Distributions

Main article: Distribution

A distribution (or generalized function) is a linear map assigning a number to each “test” function, typically a smooth function with compact support, in a continuous way: in the above terminology the space of distributions is the (con- tinuous) dual of the test function space.[74] The latter space is endowed with a topology that takes into account not only f itself, but also all its higher derivatives. A standard example is the result of integrating a test function f over some domain Ω:

∫ I(f) = f(x) dx. Ω When Ω = {p}, the set consisting of a single point, this reduces to the Dirac distribution, denoted by δ, which associates to a test function f its value at the p: δ(f) = f(p). Distributions are a powerful instrument to solve differential equations. Since all standard analytic notions such as derivatives are linear, they extend naturally to the space of distributions. Therefore, the equation in question can be transferred to a distribution space, which is bigger than the underlying function space, so that more flexible methods are available for solving the equation. For example, Green’s functions and fundamental solutions are usually distributions rather than proper functions, and can then be used to find solutions of the equation with prescribed boundary conditions. The found solution can then in some cases be proven to be actually a true function, and a solution to the original equation (e.g., using the Lax–Milgram theorem, a consequence of the Riesz representation theorem).[75]

42.8.2 Fourier analysis

Main article: Fourier analysis Resolving a periodic function into a sum of trigonometric functions forms a Fourier series, a technique much used

The heat equation describes the dissipation of physical properties over time, such as the decline of the temperature of a hot body placed in a colder environment (yellow depicts colder regions than red).

in physics and engineering.[nb 13][76] The underlying vector space is usually the Hilbert space L2(0, 2π), for which the 42.9. GENERALIZATIONS 285 functions sin mx and cos mx (m an integer) form an orthogonal basis.[77] The Fourier expansion of an L2 function f is

∞ a ∑ 0 + [a cos (mx) + b sin (mx)] . 2 m m m=1 The coefficients am and bm are called Fourier coefficients of f, and are calculated by the formulas[78] ∫ ∫ 1 2π 1 2π am = π 0 f(t) cos(mt) dt , bm = π 0 f(t) sin(mt) dt.

In physical terms the function is represented as a superposition of sine waves and the coefficients give information about the function’s frequency spectrum.[79] A complex-number form of Fourier series is also commonly used.[78] The concrete formulae above are consequences of a more general mathematical duality called Pontryagin duality.[80] Applied to the group R, it yields the classical Fourier transform; an application in physics are reciprocal lattices, where the underlying group is a finite-dimensional real vector space endowed with the additional datum of a lattice encoding positions of atoms in crystals.[81] Fourier series are used to solve boundary value problems in partial differential equations.[82] In 1822, Fourier first used this technique to solve the heat equation.[83] A discrete version of the Fourier series can be used in sampling applications where the function value is known only at a finite number of equally spaced points. In this case the Fourier series is finite and its value is equal to the sampled values at all points.[84] The set of coefficients is known as the discrete Fourier transform (DFT) of the given sample sequence. The DFT is one of the key tools of digital signal processing, a field whose applications include radar, speech encoding, image compression.[85] The JPEG image format is an application of the closely related discrete cosine transform.[86] The fast Fourier transform is an algorithm for rapidly computing the discrete Fourier transform.[87] It is used not only for calculating the Fourier coefficients but, using the convolution theorem, also for computing the convolution of two finite sequences.[88] They in turn are applied in digital filters[89] and as a rapid multiplication algorithm for polynomials and large integers (Schönhage-Strassen algorithm).[90][91]

42.8.3 Differential geometry

Main article: Tangent space The tangent plane to a surface at a point is naturally a vector space whose origin is identified with the point of contact. The tangent plane is the best linear approximation, or linearization, of a surface at a point.[nb 14] Even in a three-dimensional Euclidean space, there is typically no natural way to prescribe a basis of the tangent plane, and so it is conceived of as an abstract vector space rather than a real coordinate space. The tangent space is the generalization to higher-dimensional differentiable manifolds.[92] Riemannian manifolds are manifolds whose tangent spaces are endowed with a suitable inner product.[93] Derived therefrom, the Riemann curvature tensor encodes all curvatures of a manifold in one object, which finds applications in general relativity, for example, where the Einstein curvature tensor describes the matter and energy content of space-time.[94][95] The tangent space of a Lie group can be given naturally the structure of a Lie algebra and can be used to classify compact Lie groups.[96]

42.9 Generalizations

42.9.1 Vector bundles

Main articles: Vector bundle and Tangent bundle A vector bundle is a family of vector spaces parametrized continuously by a topological space X.[92] More precisely, a vector bundle over X is a topological space E equipped with a continuous map

π : E → X such that for every x in X, the fiber π−1(x) is a vector space. The case dim V = 1 is called a line bundle. For any vector space V, the projection X × V → X makes the product X × V into a “trivial” vector bundle. Vector bundles over X 286 CHAPTER 42. VECTOR SPACE

The tangent space to the 2-sphere at some point is the infinite plane touching the sphere in this point.

are required to be locally a product of X and some (fixed) vector space V: for every x in X, there is a neighborhood U of x such that the restriction of π to π−1(U) is isomorphic[nb 15] to the trivial bundle U × V → U. Despite their locally trivial character, vector bundles may (depending on the shape of the underlying space X) be “twisted” in the large (i.e., the bundle need not be (globally isomorphic to) the trivial bundle X × V). For example, the Möbius strip can be seen as a line bundle over the circle S1 (by identifying open intervals with the real line). It is, however, different from the cylinder S1 × R, because the latter is orientable whereas the former is not.[97] Properties of certain vector bundles provide information about the underlying topological space. For example, the tangent bundle consists of the collection of tangent spaces parametrized by the points of a differentiable manifold. The tangent bundle of the circle S1 is globally isomorphic to S1 × R, since there is a global nonzero vector field on S1.[nb 16] In contrast, by the hairy ball theorem, there is no (tangent) vector field on the 2-sphere S2 which is everywhere nonzero.[98] K-theory studies the isomorphism classes of all vector bundles over some topological space.[99] In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real division algebras: R, C, the quaternions H and the octonions. The cotangent bundle of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the cotangent space. Sections of that bundle are known as differential one-forms.

42.9.2 Modules

Main article: Module

Modules are to rings what vector spaces are to fields. The very same axioms, applied to a ring R instead of a field F yield modules.[100] The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have multiplicative inverses. For example, modules need not have bases, as the Z-module (i.e., abelian group) Z/2Z shows; those modules that do (including all vector spaces) are known as free modules. Nevertheless, a vector space can be compactly defined as a module over a ring which is a field with the elements being called vectors. Some authors use the term vector space to mean modules over a division ring.[101] The algebro- geometric interpretation of commutative rings via their spectrum allows the development of concepts such as locally free modules, the algebraic counterpart to vector bundles. 42.9. GENERALIZATIONS 287

1 S

U

1 U x R U

A Möbius strip. Locally, it looks like U × R.

42.9.3 Affine and projective spaces

Main articles: Affine space and Projective space Roughly, affine spaces are vector spaces whose origins are not specified.[102] More precisely, an affine space is a set with a free transitive vector space action. In particular, a vector space is an affine space over itself, by the map

V × V → V,(v, a) ↦ a + v.

If W is a vector space, then an affine subspace is a subset of W obtained by translating a linear subspace V by a fixed vector x ∈ W; this space is denoted by x + V (it is a coset of V in W) and consists of all vectors of the form x + v for v ∈ V. An important example is the space of solutions of a system of inhomogeneous linear equations

Ax = b

generalizing the homogeneous case b = 0 above.[103] The space of solutions is the affine subspace x + V where x is a particular solution of the equation, and V is the space of solutions of the homogeneous equation (the nullspace of A). The set of one-dimensional subspaces of a fixed finite-dimensional vector space V is known as projective space; it may be used to formalize the idea of parallel lines intersecting at infinity.[104] Grassmannians and flag manifolds generalize this by parametrizing linear subspaces of fixed dimension k and flags of subspaces, respectively. 288 CHAPTER 42. VECTOR SPACE

An affine plane (light blue) in R3. It is a two-dimensional subspace shifted by a vector x (red).

42.10 See also

• Vector (mathematics and physics), for a list of various kinds of vectors

42.11 Notes

[1] It is also common, especially in physics, to denote vectors with an arrow on top: ⃗v .

[2] This axiom refers to two different operations: scalar multiplication: bv; and field multiplication: ab. It does not assert the associativity of either operation. More formally, scalar multiplication is the semigroup action of the scalars on the vector space. Combined with the axiom of the identity element of scalar multiplication, it is a monoid action.

[3] Some authors (such as Brown 1991) restrict attention to the fields R or C, but most of the theory is unchanged for an arbitrary field.

[4] The indicator functions of intervals (of which there are infinitely many) are linearly independent, for example.

[5] The nomenclature derives from German "eigen", which means own or proper.

[6] Roman 2005, ch. 8, p. 140. See also Jordan–Chevalley decomposition.

[7] Some authors (such as Roman 2005) choose to start with this equivalence relation and derive the concrete shape of V/W from this.

[8] This requirement implies that the topology gives rise to a uniform structure, Bourbaki 1989, ch. II

[9] The triangle inequality for |−|p is provided by the Minkowski inequality. For technical reasons, in the context of functions one has to identify functions that agree almost everywhere to get a norm, and not only a seminorm. 42.12. FOOTNOTES 289

[10] “Many functions in L2 of Lebesgue measure, being unbounded, cannot be integrated with the classical Riemann integral. So spaces of Riemann integrable functions would not be complete in the L2 norm, and the orthogonal decomposition would not apply to them. This shows one of the advantages of Lebesgue integration.”, Dudley 1989, §5.3, p. 125

[11] For p ≠2, Lp(Ω) is not a Hilbert space.

[12] A basis of a Hilbert space is not the same thing as a basis in the sense of linear algebra above. For distinction, the latter is then called a Hamel basis.

[13] Although the Fourier series is periodic, the technique can be applied to any L2 function on an interval by considering the function to be continued periodically outside the interval. See Kreyszig 1988, p. 601

[14] That is to say (BSE-3 2001), the plane passing through the point of contact P such that the distance from a point P1 on the surface to the plane is infinitesimally small compared to the distance from P1 to P in the limit as P1 approaches P along the surface.

[15] That is, there is a homeomorphism from π−1(U) to V × U which restricts to linear isomorphisms between fibers.

[16] A line bundle, such as the tangent bundle of S1 is trivial if and only if there is a section that vanishes nowhere, see Husemoller 1994, Corollary 8.3. The sections of the tangent bundle are just vector fields.

42.12 Footnotes

[1] Roman 2005, ch. 1, p. 27

[2] van der Waerden 1993, Ch. 19

[3] Bourbaki 1998, §II.1.1. Bourbaki calls the group homomorphisms f(a) homotheties.

[4] Bourbaki 1969, ch. “Algèbre linéaire et algèbre multilinéaire”, pp. 78–91

[5] Bolzano 1804

[6] Möbius 1827

[7] Hamilton 1853

[8] Grassmann 2000

[9] Peano 1888, ch. IX

[10] Banach 1922

[11] Dorier 1995, Moore 1995

[12] Lang 1987, ch. I.1

[13] Lang 2002, ch. V.1

[14] e.g. Lang 1993, ch. XII.3., p. 335

[15] Lang 1987, ch. IX.1

[16] Lang 1987, ch. VI.3.

[17] Roman 2005, Theorem 1.9, p. 43

[18] Blass 1984

[19] Halpern 1966, pp. 670–673

[20] Artin 1991, Theorem 3.3.13

[21] Braun 1993, Th. 3.4.5, p. 291

[22] Stewart 1975, Proposition 4.3, p. 52

[23] Stewart 1975, Theorem 6.5, p. 74

[24] Roman 2005, ch. 2, p. 45 290 CHAPTER 42. VECTOR SPACE

[25] Lang 1987, ch. IV.4, Corollary, p. 106

[26] Lang 1987, Example IV.2.6

[27] Lang 1987, ch. VI.6

[28] Halmos 1974, p. 28, Ex. 9

[29] Lang 1987, Theorem IV.2.1, p. 95

[30] Roman 2005, Th. 2.5 and 2.6, p. 49

[31] Lang 1987, ch. V.1

[32] Lang 1987, ch. V.3., Corollary, p. 106

[33] Lang 1987, Theorem VII.9.8, p. 198

[34] Roman 2005, ch. 8, p. 135–156

[35] Lang 1987, ch. IX.4

[36] Roman 2005, ch. 1, p. 29

[37] Roman 2005, ch. 1, p. 35

[38] Roman 2005, ch. 3, p. 64

[39] Lang 1987, ch. IV.3.

[40] Roman 2005, ch. 2, p. 48

[41] Mac Lane 1998

[42] Roman 2005, ch. 1, pp. 31–32

[43] Lang 2002, ch. XVI.1

[44] Roman 2005, Th. 14.3. See also Yoneda lemma.

[45] Schaefer & Wolff 1999, pp. 204–205

[46] Bourbaki 2004, ch. 2, p. 48

[47] Roman 2005, ch. 9

[48] Naber 2003, ch. 1.2

[49] Treves 1967

[50] Bourbaki 1987

[51] Kreyszig 1989, §4.11-5

[52] Kreyszig 1989, §1.5-5

[53] Choquet 1966, Proposition III.7.2

[54] Treves 1967, p. 34–36

[55] Lang 1983, Cor. 4.1.2, p. 69

[56] Treves 1967, ch. 11

[57] Treves 1967, Theorem 11.2, p. 102

[58] Evans 1998, ch. 5

[59] Treves 1967, ch. 12

[60] Dennery 1996, p.190

[61] Lang 1993, Th. XIII.6, p. 349

[62] Lang 1993, Th. III.1.1 42.12. FOOTNOTES 291

[63] Choquet 1966, Lemma III.16.11

[64] Kreyszig 1999, Chapter 11

[65] Griffiths 1995, Chapter 1

[66] Lang 1993, ch. XVII.3

[67] Lang 2002, ch. III.1, p. 121

[68] Eisenbud 1995, ch. 1.6

[69] Varadarajan 1974

[70] Lang 2002, ch. XVI.7

[71] Lang 2002, ch. XVI.8

[72] Luenberger 1997, §7.13

[73] See representation theory and group representation.

[74] Lang 1993, Ch. XI.1

[75] Evans 1998, Th. 6.2.1

[76] Folland 1992, p. 349 ff

[77] Gasquet & Witomski 1999, p. 150

[78] Gasquet & Witomski 1999, §4.5

[79] Gasquet & Witomski 1999, p. 57

[80] Loomis 1953, Ch. VII

[81] Ashcroft & Mermin 1976, Ch. 5

[82] Kreyszig 1988, p. 667

[83] Fourier 1822

[84] Gasquet & Witomski 1999, p. 67

[85] Ifeachor & Jervis 2002, pp. 3–4, 11

[86] Wallace Feb 1992

[87] Ifeachor & Jervis 2002, p. 132

[88] Gasquet & Witomski 1999, §10.2

[89] Ifeachor & Jervis 2002, pp. 307–310

[90] Gasquet & Witomski 1999, §10.3

[91] Schönhage & Strassen 1971

[92] Spivak 1999, ch. 3

[93] Jost 2005. See also Lorentzian manifold.

[94] Misner, Thorne & Wheeler 1973, ch. 1.8.7, p. 222 and ch. 2.13.5, p. 325

[95] Jost 2005, ch. 3.1

[96] Varadarajan 1974, ch. 4.3, Theorem 4.3.27

[97] Kreyszig 1991, §34, p. 108

[98] Eisenberg & Guy 1979

[99] Atiyah 1989

[100] Artin 1991, ch. 12 292 CHAPTER 42. VECTOR SPACE

[101] Grillet, Pierre Antoine. Abstract algebra. Vol. 242. Springer Science & Business Media, 2007.

[102] Meyer 2000, Example 5.13.5, p. 436

[103] Meyer 2000, Exercise 5.13.15–17, p. 442

[104] Coxeter 1987

42.13 References

42.13.1 Algebra

• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1

• Blass, Andreas (1984), “Existence of bases implies the axiom of choice”, Axiomatic set theory (Boulder, Col- orado, 1983), Contemporary Mathematics 31, Providence, R.I.: American Mathematical Society, pp. 31–33, MR 763890

• Brown, William A. (1991), Matrices and vector spaces, New York: M. Dekker, ISBN 978-0-8247-8419-5

• Lang, Serge (1987), Linear algebra, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96412-6

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer- Verlag, ISBN 978-0-387-95385-4, MR 1878556

• Mac Lane, Saunders (1999), Algebra (3rd ed.), pp. 193–222, ISBN 0-8218-1646-2

• Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 978-0-89871-454-8

• Roman, Steven (2005), Advanced Linear Algebra, Graduate Texts in Mathematics 135 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-24766-3

• Spindler, Karlheinz (1993), Abstract Algebra with Applications: Volume 1: Vector spaces and groups, CRC, ISBN 978-0-8247-9144-5

• van der Waerden, Bartel Leendert (1993), Algebra (in German) (9th ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-56799-8

42.13.2 Analysis

• Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin, New York: Springer- Verlag, ISBN 978-3-540-13627-9

• Bourbaki, Nicolas (2004), Integration I, Berlin, New York: Springer-Verlag, ISBN 978-3-540-41129-1

• Braun, Martin (1993), Differential equations and their applications: an introduction to applied mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-97894-9

• BSE-3 (2001), “Tangent plane”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978- 1-55608-010-4

• Choquet, Gustave (1966), Topology, Boston, MA: Academic Press

• Dennery, Philippe; Krzywicki, Andre (1996), Mathematics for Physicists, Courier Dover Publications, ISBN 978-0-486-69193-0

• Dudley, Richard M. (1989), Real analysis and probability, The Wadsworth & Brooks/Cole Mathematics Series, Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, ISBN 978-0-534-10050-6

• Dunham, William (2005), The Calculus Gallery, Princeton University Press, ISBN 978-0-691-09565-3

• Evans, Lawrence C. (1998), Partial differential equations, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0772-9 42.13. REFERENCES 293

• Folland, Gerald B. (1992), Fourier Analysis and Its Applications, Brooks-Cole, ISBN 978-0-534-17094-3

• Gasquet, Claude; Witomski, Patrick (1999), Fourier Analysis and Applications: Filtering, Numerical Compu- tation, Wavelets, Texts in Applied Mathematics, New York: Springer-Verlag, ISBN 0-387-98485-2

• Ifeachor, Emmanuel C.; Jervis, Barrie W. (2001), Digital Signal Processing: A Practical Approach (2nd ed.), Harlow, Essex, England: Prentice-Hall (published 2002), ISBN 0-201-59619-9

• Krantz, Steven G. (1999), A Panorama of Harmonic Analysis, Carus Mathematical Monographs, Washington, DC: Mathematical Association of America, ISBN 0-88385-031-1

• Kreyszig, Erwin (1988), Advanced Engineering Mathematics (6th ed.), New York: John Wiley & Sons, ISBN 0-471-85824-2

• Kreyszig, Erwin (1989), Introductory functional analysis with applications, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-50459-7, MR 992618

• Lang, Serge (1983), Real analysis, Addison-Wesley, ISBN 978-0-201-14179-5

• Lang, Serge (1993), Real and functional analysis, Berlin, New York: Springer-Verlag, ISBN 978-0-387- 94001-4

• Loomis, Lynn H. (1953), An introduction to abstract harmonic analysis, Toronto-New York–London: D. Van Nostrand Company, Inc., pp. x+190

• Schaefer, Helmut H.; Wolff, M.P. (1999), Topological vector spaces (2nd ed.), Berlin, New York: Springer- Verlag, ISBN 978-0-387-98726-2

• Treves, François (1967), Topological vector spaces, distributions and kernels, Boston, MA: Academic Press

42.13.3 Historical references

• Banach, Stefan (1922), “Sur les opérations dans les ensembles abstraits et leur application aux équations inté- grales (On operations in abstract sets and their application to integral equations)" (PDF), Fundamenta Mathe- maticae (in French) 3, ISSN 0016-2736

• Bolzano, Bernard (1804), Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry) (in German)

• Bourbaki, Nicolas (1969), Éléments d'histoire des mathématiques (Elements of history of mathematics) (in French), Paris: Hermann

• Dorier, Jean-Luc (1995), “A general outline of the genesis of vector space theory”, Historia Mathematica 22 (3): 227–261, doi:10.1006/hmat.1995.1024, MR 1347828

• Fourier, Jean Baptiste Joseph (1822), Théorie analytique de la chaleur (in French), Chez Firmin Didot, père et fils

• Grassmann, Hermann (1844), Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik (in German), O. Wigand, reprint: Hermann Grassmann. Translated by Lloyd C. Kannenberg. (2000), Kannenberg, L.C., ed., Extension Theory, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2031-5

• Hamilton, William Rowan (1853), Lectures on Quaternions, Royal Irish Academy

• Möbius, August Ferdinand (1827), Der Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behand- lung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry) (in German)

• Moore, Gregory H. (1995), “The axiomatization of linear algebra: 1875–1940”, Historia Mathematica 22 (3): 262–303, doi:10.1006/hmat.1995.1025

• Peano, Giuseppe (1888), Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva (in Italian), Turin 294 CHAPTER 42. VECTOR SPACE

42.13.4 Further references

• Ashcroft, Neil; Mermin, N. David (1976), Solid State Physics, Toronto: Thomson Learning, ISBN 978-0-03- 083993-1

• Atiyah, Michael Francis (1989), K-theory, Advanced Book Classics (2nd ed.), Addison-Wesley, ISBN 978-0- 201-09394-0, MR 1043170

• Bourbaki, Nicolas (1998), Elements of Mathematics : Algebra I Chapters 1-3, Berlin, New York: Springer- Verlag, ISBN 978-3-540-64243-5

• Bourbaki, Nicolas (1989), General Topology. Chapters 1-4, Berlin, New York: Springer-Verlag, ISBN 978-3- 540-64241-1

• Coxeter, Harold Scott MacDonald (1987), Projective Geometry (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96532-1

• Eisenberg, Murray; Guy, Robert (1979), “A proof of the hairy ball theorem”, The American Mathematical Monthly (Mathematical Association of America) 86 (7): 572–574, doi:10.2307/2320587, JSTOR 2320587

• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics 150, Berlin, New York: Springer- Verlag, ISBN 978-0-387-94269-8, MR 1322960

• Goldrei, Derek (1996), Classic Set Theory: A guided independent study (1st ed.), London: Chapman and Hall, ISBN 0-412-60610-0

• Griffiths, David J. (1995), Introduction to Quantum Mechanics, Upper Saddle River, NJ: Prentice Hall, ISBN 0-13-124405-1

• Halmos, Paul R. (1974), Finite-dimensional vector spaces, Berlin, New York: Springer-Verlag, ISBN 978-0- 387-90093-3

• Halpern, James D. (Jun 1966), “Bases in Vector Spaces and the Axiom of Choice”, Proceedings of the Ameri- can Mathematical Society (American Mathematical Society) 17 (3): 670–673, doi:10.2307/2035388, JSTOR 2035388

• Husemoller, Dale (1994), Fibre Bundles (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387- 94087-8

• Jost, Jürgen (2005), Riemannian Geometry and Geometric Analysis (4th ed.), Berlin, New York: Springer- Verlag, ISBN 978-3-540-25907-7

• Kreyszig, Erwin (1991), Differential geometry, New York: Dover Publications, pp. xiv+352, ISBN 978-0-486- 66721-8

• Kreyszig, Erwin (1999), Advanced Engineering Mathematics (8th ed.), New York: John Wiley & Sons, ISBN 0-471-15496-2

• Luenberger, David (1997), Optimization by vector space methods, New York: John Wiley & Sons, ISBN 978- 0-471-18117-0

• Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer- Verlag, ISBN 978-0-387-98403-2

• Misner, Charles W.; Thorne, Kip; Wheeler, John Archibald (1973), Gravitation, W. H. Freeman, ISBN 978- 0-7167-0344-0

• Naber, Gregory L. (2003), The geometry of Minkowski spacetime, New York: Dover Publications, ISBN 978- 0-486-43235-9, MR 2044239

• Schönhage, A.; Strassen, Volker (1971), “Schnelle Multiplikation großer Zahlen (Fast multiplication of big numbers)" (PDF), Computing (in German) 7: 281–292, doi:10.1007/bf02242355, ISSN 0010-485X

• Spivak, Michael (1999), A Comprehensive Introduction to Differential Geometry (Volume Two), Houston, TX: Publish or Perish 42.14. EXTERNAL LINKS 295

• Stewart, Ian (1975), Galois Theory, Chapman and Hall Mathematics Series, London: Chapman and Hall, ISBN 0-412-10800-3 • Varadarajan, V. S. (1974), Lie groups, Lie algebras, and their representations, Prentice Hall, ISBN 978-0-13- 535732-3 • Wallace, G.K. (Feb 1992), “The JPEG still picture compression standard”, IEEE Transactions on Consumer Electronics 38 (1): xviii–xxxiv, doi:10.1109/30.125072, ISSN 0098-3063 • Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathe- matics 38, Cambridge University Press, ISBN 978-0-521-55987-4, OCLC 36131259, MR 1269324

42.14 External links

• Hazewinkel, Michiel, ed. (2001), “Vector space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608- 010-4

• A lecture about fundamental concepts related to vector spaces (given at MIT) • A graphical simulator for the concepts of span, linear dependency, base and dimension 296 CHAPTER 42. VECTOR SPACE

42.15 Text and image sources, contributors, and licenses

42.15.1 Text • 2 × 2 real matrices Source: https://en.wikipedia.org/wiki/2_%C3%97_2_real_matrices?oldid=660100486 Contributors: Gareth Owen, Toby Bartels, Michael Hardy, TakuyaMurata, Giftlite, BenFrantzDale, Rgdboer, Jheald, SmackBot, Incnis Mrsi, Lambiam, Jim.belk, STBot, It Is Me Here, Haseldon, DavidCBryant, Geometry guy, Anchor Link Bot, Sun Creator, Marc van Leeuwen, TutterMouse, Yobot, AnomieBOT, BG19bot, Mark L MacDonald, Loraof and Anonymous: 12 • 3D projection Source: https://en.wikipedia.org/wiki/3D_projection?oldid=686294182 Contributors: Mrwojo, Patrick, Michael Hardy, Flamurai, Alfio, Cyp, Angela, Raven in Orbit, Schneelocke, Charles Matthews, Sboehringer, Furrykef, Omegatron, Bloodshedder, Robbot, Jaredwf, Altenmann, RossA, Cholling, Giftlite, BenFrantzDale, Dratman, Tristanreid, Ctachme, Allefant, Aniboy2000, RJHall, Sietse Snel, Marco Polo, Pearle, Mdd, Stephan Leeds, Oleg Alexandrov, Zanaq, Lincher, Waldir, Plowboylifestyle, Que, Rjwilmsi, Kevmitch, Mathbot, Krishnavedala, Bgwhite, Froth, MathsIsFun, SmackBot, Canthusus, Chris the speller, Bluebot, TimBentley, Nbarth, Tamfang, Berland, PsychoAlienDog, Ckatz, Van helsing, Harej bot, Michaelbarreto, Luckyherb, Davidhorman, Bobbygao, PhilKnight, Deom, Seet82, Cpl Syx, Akulo, R'n'B, SharkD, RenniePet, Remi0o, Technopat, Paolo.dL, Dhatfield, Stestagg, Martarius, ClueBot, Gamer Eek, Speshall, Tyler, Skytiger2, Sandeman684, Baudway, Addbot, LaaknorBot, Yobot, Datadelay~enwiki, Rabiee, Citation bot, Akilaa, Vgergo, TinucherianBot II, Unigfjkl, BrainFRZ, Miym, FrescoBot, Citation bot 1, Bunyk, Drag0nius, Trappist the monk, RjwilmsiBot, John of Reading, Pickypickywiki, Jovianconflict, Heymid, ZéroBot, AManWithNoPlan, Ego White Tray, ClueBot NG, Twillisjr, Widr, Wiki13, Aekquy, Ieay4a, Kees3125, Janelbriosyk and Anonymous: 102 • Affine coordinate system Source: https://en.wikipedia.org/wiki/Affine_coordinate_system?oldid=560477276 Contributors: Mathbot, Incnis Mrsi, David Eppstein, Addbot, AnomieBOT, Xqbot and EmausBot • Affine geometry Source: https://en.wikipedia.org/wiki/Affine_geometry?oldid=686235369 Contributors: Zundark, Toby Bartels, Si- monP, Patrick, Michael Hardy, Rp, TakuyaMurata, Poor Yorick, Schneelocke, Charles Matthews, Dysprosia, Gutsul, Phys, Topbanana, Skeetch, Henrygb, Bkell, Marc Venot, Tosha, Giftlite, SteffenB~enwiki, D6, Bender235, Zaslav, Rgdboer, Cje~enwiki, Msh210, Dirac1933, Ceyockey, Oleg Alexandrov, Woohookitty, Mathbot, Siddhant, YurikBot, Michael Slone, Tong~enwiki, Długosz, Yoosef Pooranvary, Wangi, Cjfsyntropy, Bluebot, DHN-bot~enwiki, Anthon.Eff, Vaughan Pratt, Simian1k, Phatom87, Ntsimp, Thijs!bot, Konradek, MattWatt, The Transhumanist, David Eppstein, MartinBot, Schmloof, Terrek, TomyDuby, Redcoral, VolkovBot, JohnBlackburne, Rei-bot, Brianga, Radon210, La fee cerule, Svick, Justin W Smith, Marino-slo, Marc van Leeuwen, Addbot, Zorrobot, Legobot, Luckas-bot, 9258fahs- flkh917fas, Crystal whacker, Citation bot, Omnipaedista, MorphismOfDoom, SUL, RjwilmsiBot, EmausBot, Quondum, D.Lazard, Brad7777, Khazar2, K9re11, WillemienH, Loraof, KasparBot and Anonymous: 30 • Affine group Source: https://en.wikipedia.org/wiki/Affine_group?oldid=678086830 Contributors: AxelBoldt, Patrick, Michael Hardy, Docu, Schneelocke, Loren Rosen, Charles Matthews, Dysprosia, Sixpence, Rgdboer, Jheald, Ceyockey, Bgwhite, Cullinane, Smack- Bot, Nbarth, Vaughan Pratt, Dharma6662000, Thijs!bot, Turlo Lomon, Magioladitis, Vanish2, David Eppstein, Maproom, JackSchmidt, Alexbot, Addbot, Luckas-bot, Yobot, DSisyphBot, Orenburg1, MaximalIdeal, Svetivlas and Anonymous: 10 • Affine space Source: https://en.wikipedia.org/wiki/Affine_space?oldid=687756487 Contributors: Toby Bartels, Edward, Patrick, Michael Hardy, TakuyaMurata, Schneelocke, Charles Matthews, Dysprosia, Gandalf61, Robbar~enwiki, Tosha, Giftlite, BenFrantzDale, Lethe, SteffenB~enwiki, DemonThing, Vadmium, Paul August, Elwikipedista~enwiki, Rgdboer, EmilJ, Tsirel, Msh210, Eric Kvaalen, Oleg Alexandrov, Joriki, BD2412, MarSch, Salix alba, R.e.b., Chobot, YurikBot, Wavelength, Wolfmankurd, Archelon, RFBailey, Crasshop- per, Sir Dagon, Netrapt, Mebden, Sdayal, SmackBot, Incnis Mrsi, Optikos, Silly rabbit, Nbarth, Richard L. Peterson, Tyrrell McAllis- ter, Rschwieb, Newone, Zero sharp, CmdrObot, CBM, Shreyasjoshis, Myasuda, Headbomb, Flarity, VictorAnyakin, JAnDbot, Rogier- Brussee, Albmont, Gulloar, TomyDuby, Marcosaedro, Swagato Barman Roy, VanishedUserABC, Paolo.dL, OKBot, Anchor Link Bot, Marcus.bishop, Justin W Smith, Bpavel88, Bender2k14, Hans Adler, SchreiberBike, Silasdavis, Beroal, RQG, Addbot, TheGeekHead, Jarble, Luckas-bot, Yobot, AnomieBOT, Ziyuang, Citation bot, ArthurBot, DannyAsher, Quebec99, FrescoBot, Sławomir Biały, Cita- tion bot 1, SUL, EmausBot, Slawekb, Quondum, D.Lazard, Jeroendv, ChuispastonBot, Makhokh, Jj1236, Mgvongoeden, Mesoderm, TylerWRoss, Helpful Pixie Bot, Mijagourlay, JellyPatotie, ObviouslyNotASock and Anonymous: 67 • Affine transformation Source: https://en.wikipedia.org/wiki/Affine_transformation?oldid=685926001 Contributors: Damian Yerrick, AxelBoldt, SimonP, Jdlh, Patrick, Chas zzz brown, Michael Hardy, J-Wiki, CatherineMunro, Schneelocke, Charles Matthews, Dcoetzee, Dysprosia, Jitse Niesen, Hyacinth, Phys, Skeetch, Robbot, Josh Cherry, Fredrik, Altenmann, Guy Peters, Giftlite, Gene Ward Smith, BenFrantzDale, MSGJ, Ajgorhoe, LucasVB, Mysidia, Felix Wiemann, Discospinster, Guanabot, Dave Foley, Rgdboer, EmilJ, Longhair, .:Ajvol:., Foobaz, Dvgrn, Jumbuck, Jheald, Alaudo, Oleg Alexandrov, LOL, Mpatel, BD2412, FlaBot, Mathbot, Margosbot~enwiki, Kri, YurikBot, Michael Slone, Hellbus, Archelon, Tong~enwiki, Trovatore, LeonardoRob0t, Curpsbot-unicodify, Cjfsyntropy, Cmglee, SmackBot, Mmernex, Incnis Mrsi, KocjoBot~enwiki, Gilliam, IvanAndreevich, DHN-bot~enwiki, Mohan1986, Addshore, Yoshigev, Eliyak, Breno, Jimmy Pitt, Mets501, Iridescent, PavelCurtis, Agreeney, Ezrakilty, Cydebot, Mikewax, Urdutext, Mhaitham.shammaa, Battaglia01, Albmont, Rich257, JoergenB, R'n'B, Lantonov, Nwbeeson, Policron, DavidCBryant, PerezTerron, Pleasantville, Alnok- taBOT, Softtest123, Quietbritishjim, Paolo.dL, SanderEvers, Iandiver, JimInTheUSA, Lord Bruyanovich Bruyanov, Cacadril, Nostrati- cispeak, Templarion, Marc van Leeuwen, Addbot, TheGeekHead, Cesiumfrog, Zorrobot, Luckas-bot, Yobot, Ht686rg90, AnomieBOT, Quebec99, Xqbot, Omnipaedista, VadimIppolitov, Sławomir Biały, Flinx, Smutny, EmausBot, ZéroBot, Quondum, Wikfr, Erget2005, Anita5192, Mgvongoeden, Mesoderm, JornAnke, Helpful Pixie Bot, Jerrydeanrsmith, Burningstarfour, Nishch, Brad7777, BattyBot, Hierarchivist, Sylvanarevalo, AndyThe, Ebag7125, Leegrc, Apapsis and Anonymous: 81 • Augmented matrix Source: https://en.wikipedia.org/wiki/Augmented_matrix?oldid=670245141 Contributors: Giftlite, Alison, Mh, El C, 3mta3, Caleb666, Oleg Alexandrov, Bjones, StradivariusTV, GrundyCamellia, Duomillia, 48v, BiH, Octahedron80, Addshore, Shad- owdragon07, AdrianX, Mwhiz, Vanish2, Carlicus~enwiki, Sarregouset, Randomblue, Cliff, Truthnlove, Addbot, Cabwood, Luckas-bot, GrouchoBot, Nageh, Husoski, Duoduoduo, Slawekb, QEDK, ClueBot NG, Bazuz, Qetuth, YFdyh-bot, Makecat-bot and Anonymous: 24 • Barycenter Source: https://en.wikipedia.org/wiki/Barycenter?oldid=686974433 Contributors: Bryan Derksen, Tarquin, Patrick, JohnOwens, Michael Hardy, Gabriel, Charles Matthews, The Anomebot, 1984, Meelar, Pko, Herbee, Joe Kress, Beatnick~enwiki, Karol Langner, Icairns, A2Kafir, Saperaud~enwiki, Brighterorange, Vuong Ngan Ha, Zhatt, Haoie, Dbfirs, Melchoir, J 1982, JorisvS, George100, Eric, RockMFR, Sanya3, Rreagan007, V35b, Paine Ellsworth, Duoduoduo, Wcherowi, Glevum, Rfassbind, Evensteven, MaxCasey1, StressOverStrain and Anonymous: 13 • Bent function Source: https://en.wikipedia.org/wiki/Bent_function?oldid=678111371 Contributors: Phil Boswell, Rich Farmbrough, Will Orrick, Rjwilmsi, Marozols, Ntsimp, David Eppstein, Anonymous Dissident, Watchduck, AdmiralHood, Nageh, Citation bot 1, 42.15. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 297

Trappist the monk, Gzorg, EmausBot, Wcherowi, Helpful Pixie Bot, ChrisGualtieri, Zieglerk, Monkbot, Cryptowarrior and Anonymous: 3 • Cartesian coordinate system Source: https://en.wikipedia.org/wiki/Cartesian_coordinate_system?oldid=687737432 Contributors: Damian Yerrick, Chuck Smith, Bryan Derksen, Zundark, Tarquin, Mark Ryan, Andre Engels, Heron, Montrealais, Patrick, Michael Hardy, Dcljr, JWSchmidt, Александър, AugPi, Skyfaller, Smack, Pizza Puzzle, Nikola Smolenski, Drz~enwiki, Emperorbma, Charles Matthews, Timwi, Dysprosia, Colipon, The Anomebot, Maximus Rex, Bevo, Chuunen Baka, Robbot, Romanm, Modulatum, Sverdrup, Henrygb, Prara, Michael Snow, Jleedev, Enochlau, Snobot, Giftlite, DocWatson42, Jason Quinn, Jorge Stolfi, Manuel Anastácio, SoWhy, Mr- Mambo, Joseph Myers, C4~enwiki, Adashiel, Discospinster, Rich Farmbrough, Guanabot, Paul August, Bender235, Elwikipedista~enwiki, BenjBot, Edwinstearns, Rgdboer, Mavhc, Che090572, Kjkolb, Larry V, HasharBot~enwiki, OGoncho, Jumbuck, Alansohn, Snow- Fire, PAR, Yossiea~enwiki, Fordan, Dionoea, Kbolino, Falcorian, Oleg Alexandrov, Mel Etitis, Woohookitty, Linas, LOL, Firefishy, Mpatel, Waldir, Ruziklan, Palica, Tony1849, Graham87, Jobnikon, BD2412, Rjwilmsi, MJSkia1, Ygrek, OneWeirdDude, MarSch, Jmcc150, Salix alba, Brighterorange, Titoxd, FlaBot, Mathbot, Nihiltres, Maxal, Cmbrannon, Slant, Fresheneesz, Srleffler, Chobot, Helios, DVdm, YurikBot, Wavelength, RobotE, Charles Gaudette, Pip2andahalf, RussBot, Hede2000, Piet Delport, Gustavb, Byj2000, RabidDeity, Dhollm, Bucketsofg, Dbfirs, Cheeser1, DeadEyeArrow, Dast, MathsIsFun, HereToHelp, Tony Liao~enwiki, JLaTondre, Gesslein, Archer7, Allens, Katieh5584, Meegs, Cmglee, Nekura, DVD R W, Sardanaphalus, SmackBot, RDBury, Adam majewski, Incnis Mrsi, Delldot, Canthusus, ParlorGames, Gilliam, Skizzik, Mirokado, Kurykh, Miquonranger03, Silly rabbit, Basalisk, Octahe- dron80, Nbarth, DHN-bot~enwiki, Hongooi, Antonrojo, Can't sleep, clown will eat me, RedHillian, SundarBot, Cameron Nedland, Memming, Downtown dan seattle, Doodle77, Andeggs, Sadi Carnot, Andrei Stroe, Lambiam, Dbtfz, Kusarbo, Cronholm144, Bjanku- loski06en~enwiki, Aleenf1, IronGargoyle, 041744, JHunterJ, Stwalkerster, Mets501, Wizard191, Maelor, JForget, CmdrObot, Jackzhp, Dgw, NickW557, 345Kai, WeggeBot, Yaris678, WillowW, LouisBB, ST47, Edgerck, Juansempere, DumbBOT, TJ09, Vanished User jd- ksfajlasd, Zalgo, Lanepierce, Thijs!bot, Epbr123, Headbomb, Federhalter, Escarbot, Cyclonenim, AntiVandalBot, Gioto, Mhaitham.shammaa, Modernist, JAnDbot, Chausx, PhilKnight, DeclinedShadow, Magioladitis, Connormah, VoABot II, Catslash, Mrfence, CattleGirl, Fab- rictramp, Srice13, Maniwar, David Eppstein, DerHexer, MartinBot, Rettetast, Snozzer, J.delanoy, Captain panda, Sasajid, Trusilver, Numbo3, Century0, Cpiral, Lantonov, Stan J Klimas, Samtheboy, NewEnglandYankee, Davecrosby uk, CardinalDan, RJASE1, Idioma- bot, KillerOfThem, VolkovBot, JohnBlackburne, Mun206, TXiKiBoT, Anonymous Dissident, Steven J. Anderson, Martin451, Uncar- inggunner, Domitius, Topherjasmin09, Andy Dingley, Dirkbb, Blindman.rms, AlleborgoBot, Logan, Katzmik, SieBot, Euryalus, Yintan, Joaosampaio, Flyer22 Reborn, Man It’s So Loud In Here, Paolo.dL, Oxymoron83, Atmamatma, Jurlinga, Hello71, Hobartimus, Svick, Anchor Link Bot, Randomblue, Escape Orbit, Martarius, ClueBot, The Thing That Should Not Be, Mild Bill Hiccup, DragonBot, Ex- cirial, Abrech, SockPuppetForTomruen, Thingg, Mattreedywiki, Johnuniq, Darkicebot, CaptainVideo890, Skunkboy74, Vanostran, Hy- perweb79, Addbot, Binary TSO, DougsTech, Fgnievinski, Blethering Scot, Jncraton, Fieldday-sunday, Leszek Jańczuk, MrOllie, Allliam, TheFreeloader, Tide rolls, Alanfeynman, Lightbot, Nobono9, Zorrobot, Վազգեն, Wmplayer, Wwannsda, Luckas-bot, Yobot, Fraggle81, Anypodetos, Nallimbot, Vltava 68, Tempodivalse, AnomieBOT, 1exec1, Rajmathi mehta, Piano non troppo, Ulric1313, Materialscien- tist, Citation bot, Oftopladb, Xqbot, Waffleman12, Jeffwang, NorbDigiBeaver, Almabot, Frosted14, Red van man, A.amitkumar, Fres- coBot, Appropo, Majopius, Masterknighted, Rhino bucket, Wireless Keyboard, Þjóðólfr, Pmokeefe, Jsjunkie, Rogiemac, Shanmugamp7, ,Lotje, Colin Cochrane, Dasteve, Suffusion of Yellow, Shanker Pur, NameIsRon, Timh3221, Slon02, EmausBot ,کاشف عقیل ,Rausch Acather96, WikitanvirBot, GoingBatty, RA0808, Wikipelli, Slawekb, CanonLawJunkie, Knight1993, Junelvillejo, Quondum, MonoAV, Chewings72, WMC, ClueBot NG, Gareth Griffith-Jones, KlappCK, Wcherowi, MelbourneStar, Satellizer, Movses-bot, Widr, Helpful Pixie Bot, DBigXray, Popsh, Papadim.G, Questionefisica, Mark Arsten, Blue Mist 1, Phl.jns, Nbrothers, Pbierre, Pratyya Ghosh, Sergean- tHippyZombie, Radio15dude, ChrisGualtieri, Bigloser12345loser, EuroCarGT, Ramesepirate, Ducknish, Kelvinsong, Kingbowen, Web- client101, Indiana State, RazrRekr201, Jc86035, Acetotyce, ProtossPylon, Liekturtles, SamX, Wamiq, Sicaeffect, Ginsuloft, Primalshell, UY Scuti, Stamptrader, Andreatristan, Akhilburle, Wilson Widyadhana, BlueFenixReborn, Roshmita, Elsa1098, Hdkeudhdjisjedu and Anonymous: 475 • Collinearity Source: https://en.wikipedia.org/wiki/Collinearity?oldid=687597439 Contributors: Michael Hardy, Charles Matthews, Car- lossuarez46, Bearcat, Jason Quinn, Dfrankow, Bender235, Rgdboer, PAR, Btyner, BD2412, Srleffler, TexasAndroid, Cojoco, Incnis Mrsi, Lawrencekhoo, BiT, Nbarth, Tamfang, Paul Foxworthy, Thrapper, Headbomb, Escarbot, Magioladitis, Squidonius, VolkovBot, HiDrNick, Bfpage, Paolo.dL, Rumping, DumZiBoT, Dthomsen8, Mitch Ames, Tayste, Addbot, SpBot, ThaddeusB, Xqbot, Eric Yurken, Silver Spoon Sokpop, Tkuvho, Duoduoduo, Ripchip Bot, ZéroBot, Quondum, Fanyavizuri, ClueBot NG, Wcherowi, Estopedist1, MerlIw- Bot, Chafe66, Circlesareround, Dexbot, Faizan, DavidLeighEllis, Rounaq Malhotra, Loraof, Ffilozov, Pdiggidy1, Timrosens and Anony- mous: 13 • Endianness Source: https://en.wikipedia.org/wiki/Endianness?oldid=686757609 Contributors: Damian Yerrick, AxelBoldt, Tarquin, Ed Poor, Deb, SJK, Mjb, Patrick, RTC, Nixdorf, MartinHarper, TakuyaMurata, MichaelJanich, Ootachi, BigFatBuddha, Bogdangiusca, Susurrus, Cimon Avaro, GCarty, EdH, Mxn, LordK, Ehn, Emperorbma, Tcoahran, Timwi, Dcoetzee, Dysprosia, Andrewman327, Doradus, Greenrd, Tb, Zoicon5, Furrykef, Grendelkhan, Val42, LMB, Omegatron, Wernher, Pakaran, Rossh, Murray Langton, Low- ellian, P0lyglut, Pingveno, Blainster, Alba, Casito, Jleedev, Pengo, Tobias Bergemann, Dbenbenn, Smjg, DocWatson42, BenFrantzDale, Markus Kuhn, Mellum, Cpk, AJim, Langec, Chrismear, Jason Quinn, Mboverload, Rchandra, Nayuki, Ojl, Gustnado, Jastrow, Vad- mium, Mike R, Zarkonnen, HorsePunchKid, Joeblakesley, Kusunose, Maximaximax, Simoneau, Joyous!, MementoVivere, Kmccoy, Mormegil, Richie, RossPatterson, Pixel8, Lulu of the Lotus-Eaters, Roo72, Alistair1978, Bender235, Kjoonlee, Plugwash, Shanes, Jp- gordon, Alxndr, Overmann, Army1987, R. S. Shaw, Mww42, Johnteslade, Dee Earley, Greenleaf~enwiki, Chbarts, Trevj, Zr40, Helix84, Eje211, Anthony Appleyard, Guy Harris, Nealcardwell, BobGibson~enwiki, RoySmith, Miltonhowe, Benhutchings, Suruena, Danhash, Runtime, Ecn5093, Zeth, Kenyon, TShilo12, Aadnk, A D Monroe III, Jkt, Woohookitty, Shreevatsa, Tripodics, Krille, CPES, Gniw, Jimgawn, TAKASUGI Shinji, Ilya, Don Braffitt, Qwertyus, Vincent Lefèvre, Kbdank71, Teque5, Rjwilmsi, Nandesuka, Stopsineman, Husky, DirkvdM, Martinfrank, Fish and karate, Gringer, StuartBrady, FlaBot, Mirror Vax, Arnero, Musical Linguist, SDaniel, Frag- glet, Penedo, DevastatorIIC, Quuxplusone, Exelban, Fresheneesz, Micler, OpenToppedBus, Imnotminkus, Chobot, Antilived, YurikBot, Wavelength, Adrianob, Fgrieu, Hairy Dude, Deeptrivia, DMahalko, Fabartus, Pburka, Kenn, Joebeone, IanManka, Hydrargyrum, Premil, Gaius Cornelius, Anomalocaris, Długosz, Vanished user 1029384756, Aaron Brenneman, D. F. Schmidt, Lobwedge, Chewyrunt, Tony1, Karlosian, EEMIV, Falcon9x5, Jeh, Hakeem.gadi, Petr.adamek, MarkBrooks, Light current, Sqweek, CWenger, HeavyStorm, Andy- luciano~enwiki, GoodSirJava, Chr0n0ss, Jer ome, Samwilson, Ozzmosis, SmackBot, Slashme, Henriok, Unyoyega, Dandin1, Btwied, Gilliam, Lighthill, Hippodrome, Adun12, Marc321, Nbarth, BBCWatcher, Gracenotes, Kmag~enwiki, Anabus, JonHarder, Dinjiin, Dweaver, GDM, Cybercobra, Tompsci, Kukini, Davipo, HpaScalar, SashatoBot, Gennaro Prota, Salty!, Btritchie, Bzorro, JohnI, Col- lect, Dicklyon, EdC~enwiki, MTSbot~enwiki, Peyre, DagErlingSmørgrav, Kvng, Norm mit, Saviourmachine, Joseph Solis in Australia, Zero sharp, Phonetagger, DKqwerty, Barryd815, Amniarix, CmdrObot, Ivan Pozdeev, Jthill, Raysonho, HenkeB, Neelix, Tim1988, Haxial, A876, LinuxSneaker, Yeanold Viskersenn, Christian75, The Mad Bomber, Papajohnin, Thijs!bot, Falcotron, Kronos, Sobreira, 298 CHAPTER 42. VECTOR SPACE

The Wednesday Island, Jesuswaffle, Sbandrews, Malvineous, Cutullus, Esmond.pitt, JAnDbot, NapoliRoma, Albany NY, BrotherE, Fer- ritecore, Spiri82~enwiki, Destynova, Mateo2, Lenschulwitz, Bblanc, Zarathud, Harchtocin, Enquire, Eiselekd, Quanticles, R'n'B, Frankj- mattia, Uncle Dick, Alexwright, Cpiral, Silas S. Brown, RenniePet, Vfbp-geyf, Ohms law, Bigdumbdinosaur, Potatoswatter, Dorgan- Bot, Ross Fraser, Spelemann, Agamemnus, Rsaxvc, Netuser500, Ziounclesi, Yaan, Jmath666, Mratzloff, Michael Frind, TheStarman, Mrostam, Kbrose, SieBot, Caltas, Nemo20000, Jerryobject, Jimthing, Lightmouse, SimonTrew, Frappucino, ChorizoLasagna, Chieffan- cypants, Anchor Link Bot, Ed Avis, ClueBot, SatPhil, Rilak, KenShirriff, Niceguyedc, Excirial, Alexbot, Sun Creator, Kausikghatak, TobiasPersson, Aprock, Rvinjamuri, Dickguertin, Theking2, Joel Saks, Plumbum2, David Delony, Addbot, Wildplum69, Jmbpiano, Tothwolf, Netsmith2001, Ethanpet113, Scientus, Download, SpBot, AgadaUrbanit, Carlos Rosa PT, Jarble, Dennis k85, A:-)Brunuś, Legobot, Luckas-bot, Yobot, Amirobot, Caracho, EnTerr, AnomieBOT, Qdinar, Materialscientist, Slay2k, Sellyme, MarkWarren, Mart- nym, Br77rino, Kithira, Frosted14, Pandamonia, Wojtow, Vskytta, FaTony, Mhadi.afrasiabi, Bertramlyons, Kvgd, Stefan Weil, Mahnut, Faizi.qau, Daemonaka, Uberlux, Jfmantis, EmausBot, Dstarfire, Bleakgadfly, Nomen4Omen, Spuriogram, Pololei, Pejjen, Ὁ οἶστρος, H3llBot, KYLEMONGER, Zephyrus Tavvier, Donner60, Ipsign, Poster Nutbag, ClueBot NG, Jake Petroules, Matthiaspaul, Jiri 1984, Cheater no1, Hrgwea, Reify-tech, MerlIwBot, Dgordon562, Fnordly, BG19bot, Thepoliticalmaster, Eidab, Afree10, Ldaugher, Pateriam, Proxyma, Arcticaribou, ChrisGualtieri, NevonCloud, Strawberrypineapple, Mogism, SFK2, Ajacombs, Poltoratsk, Greenstruck, Tariq Noman, JPaestpreornJeolhlna, Comp.arch, Lekshmi.gk, Skibum3731, Frmkla, Monkbot, Tigercompanion25, Paratridle, Kaiser7402, OMPIRE, Adrichman, Oiyarbepsy, Vanniaz and Anonymous: 454 • Euclidean distance Source: https://en.wikipedia.org/wiki/Euclidean_distance?oldid=680026113 Contributors: Damian Yerrick, Axel- Boldt, XJaM, Boleslav Bobcik, Michael Hardy, Nikai, Epl18, AnthonyQBachler, Fredrik, Altenmann, MathMartin, Saforrest, Enochlau, Giftlite, BenFrantzDale, Bender235, Rgdboer, Bobo192, Dvogel, Obradovic Goran, Fawcett5, Oleg Alexandrov, Warbola, Ruud Koot, Isnow, Qwertyus, Unused007, Ckelloug, DVdm, Wavelength, Multichill, Number 57, StuRat, Arthur Rubin, Clams, SmackBot, Reverend- Sam, InverseHypercube, 127, Mcld, Chris the speller, Oli Filth, Papa November, Octahedron80, Nbarth, Tsca.bot, OrphanBot, Bomb- shell, Lambiam, Delfinite, Aldarione, Jminguillona, Thijs!bot, JAnDbot, .anacondabot, Theunicyclegirl, Yesitsapril, Graeme.e.smith, Robertgreer, Cometstyles, JohnBlackburne, TXiKiBoT, FrederikHertzum, Tiddly Tom, Paolo.dL, Dattorro, Justin W Smith, DragonBot, Freebit50, Triathematician, Qwfp, Zik2, Ali Esfandiari, SilvonenBot, Addbot, Fgnievinski, AkhtaBot, Tanhabot, LaaknorBot, Favonian, West.andrew.g, Yobot, Ehaussecker, Nallimbot, Ciphers, Materialscientist, Xqbot, Simeon87, Erik9bot, Gleb.svechnikov, Sławomir Bi- ały, RedBot, EmausBot, Rasim, RA0808, Cskudzu, Quondum, Kweckzilber, EdoBot, ClueBot NG, Wcherowi, Stultiwikia, Papadim.G, Ascoldcaves, Arrogantrobot, Soni, Jcarrete, ShuBraque, Erotemic, 7Sidz, Loraof and Anonymous: 70 • Euclidean space Source: https://en.wikipedia.org/wiki/Euclidean_space?oldid=681868374 Contributors: AxelBoldt, Mav, Zundark, Tarquin, XJaM, Youandme, Tomo, Patrick, Michael Hardy, Dcljr, Karada, Looxix~enwiki, Angela, Charles Matthews, Dysprosia, Gren- delkhan, David Shay, MathMartin, Tobias Bergemann, Tosha, Giftlite, Lethe, Fropuff, Sriehl, DefLog~enwiki, Andycjp, Tomruen, Iantresman, Tzanko Matev, JohnArmagh, Rich Farmbrough, Paul August, Rgdboer, Msh210, Jimmycochrane, PAR, Eddie Dealtry, Dirac1933, Woohookitty, Isnow, Qwertyus, MarSch, MZMcBride, VKokielov, Kolbasz, Fresheneesz, NevilleDNZ, Chobot, Bgwhite, JPD, Wavelength, Hede2000, Epolk, KSmrq, SpuriousQ, ENeville, Mgnbar, Arthur Rubin, Brian Tvedt, RG2, JDspeeder1, SmackBot, Iamhove, Incnis Mrsi, Reedy, Mhss, JoeKearney, Silly rabbit, Hongooi, Tamfang, SashatoBot, Jim.belk, DabMachine, Dan Gluck, Kaarel, Yggdrasil014, Heqs, CmdrObot, GargoyleMT, Rudjek, Philomath3, Aiko, Guy Macon, Orionus, Salgueiro~enwiki, JAnDbot, Bencher- lite, CrizCraig, Magioladitis, TheChard, Avicennasis, Nucleophilic, Oderbolz, R'n'B, Reedy Bot, Policron, Trigamma, The enemies of god, Cerberus0, VolkovBot, IWhisky, Philip Trueman, Richardohio, WereSpielChequers, Da Joe, Caltas, Paolo.dL, MiNombreDeGuerra, Lightmouse, Denisarona, Tomas e, Mild Bill Hiccup, Gwguffey, Vsage, DhananSekhar, SilvonenBot, SkyLined, The Rationalist, Addbot, AkhtaBot, Pmod, Tide rolls, Legobot, Yobot, , Collieuk, Materialscientist, Citation bot, Sandip90, Xqbot, St.nerol, Nfr-Maat, Dead- clever23, RoyLeban, Ksuzanne, Mineralquelle, FrescoBot, Sławomir Biały, Alxeedo, RandomDSdevel, Gapato, Mikrosam Akademija 2, Yunesj, Wikivictory, EmausBot, John of Reading, Quondum, Gbsrd, ClueBot NG, Wcherowi, Master Uegly, Cntras, Frank.manus, ElectricUvula, ElphiBot, MRG90, FeralOink, Userbot12, Lugia2453, Brirush, Limit-theorem, Eyesnore, Yardimsever, Tentinator, Fen- tonville, Mgkrupa, BemusedObserver, OrganicAltMetal, Ro4sho, Preethambittu, KasparBot, Dan6233 and Anonymous: 99 • Flat (geometry) Source: https://en.wikipedia.org/wiki/Flat_(geometry)?oldid=646971094 Contributors: Michael Hardy, Giftlite, Tom- ruen, Rgdboer, Richwales, BD2412, KSmrq, Thunderforge, Sangwine, SmackBot, Incnis Mrsi, Jim.belk, Magioladitis, David Eppstein, Infovarius, SanderEvers, Justin W Smith, Yobot, GoingBatty, Quondum, Loraof and Anonymous: 2 • Function composition Source: https://en.wikipedia.org/wiki/Function_composition?oldid=687828500 Contributors: Zundark, Tarquin, Patrick, Michael Hardy, Wshun, Kku, Dcljr, TakuyaMurata, Glenn, Andres, Charles Matthews, Timwi, Greenrd, Phys, Phil Boswell, Robbot, Rasmus Faber, Tobias Bergemann, Giftlite, Lethe, MSGJ, Dratman, Jason Quinn, Macrakis, Rheun, Karl Dickman, Paul August, Danakil, EmilJ, HasharBot~enwiki, Oleg Alexandrov, Woohookitty, Linas, Georgia guy, Zenkat, MattGiuca, Mpatel, MFH, GregorB, Qwertyus, SixWingedSeraph, Rjwilmsi, Slac, FlaBot, VKokielov, Vonkje, YurikBot, Grubber, NawlinWiki, Googl, Netrapt, Maksim- e~enwiki, Adam majewski, Incnis Mrsi, Melchoir, XudongGuan~enwiki, BiT, Nbarth, Javalenok, J•A•K, SundarBot, Ecsnp, Jon Awbrey, Lambiam, Dmh~enwiki, DA3N, Dfass, EdC~enwiki, Cherry Cotton, CBM, Strangelv, Juansempere, Escarbot, QuiteUnusual, Kuteni, JAnDbot, Gcm, Fuzzybyte, Magioladitis, David Eppstein, Policron, Cuzkatzimhut, VolkovBot, Pleasantville, JohnBlackburne, LokiClock, TXiKiBoT, Anonymous Dissident, Jonnyappleseed24, TrippingTroubadour, AlleborgoBot, EmxBot, Pit-trout, Classicalecon, ClueBot, Marino-slo, Plastikspork, SoxBot III, XLinkBot, Addbot, Lightbot, PV=nRT, Luckas-bot, Pcap, KamikazeBot, Zubachi, LilHelpa, Xqbot, RibotBOT, Einkil, Constructive editor, Pinethicket, RedBot, Fallenness, John of Reading, ZéroBot, Quondum, Aughost, EdoBot, ClueBot NG, Wcherowi, Aurelian Radoaca, Gauravjuvekar, Shashank rathore, Brad7777, ChrisGualtieri, JamesHaigh, Mathdiskteacher, Makecat- bot, Stephan Kulla, Imareaver, Jochen Burghardt, Limit-theorem, Monkbot, Ashleyelizabethmath4626, Lkmhokie8, JMP EAX, Dchsnq and Anonymous: 67 • General linear group Source: https://en.wikipedia.org/wiki/General_linear_group?oldid=676367140 Contributors: AxelBoldt, Zun- dark, Patrick, Chas zzz brown, Michael Hardy, Zhaoway~enwiki, A5, Charles Matthews, Dysprosia, Jitse Niesen, Shizhao, Huppy- banny, Weialawaga~enwiki, Giftlite, MSGJ, Fropuff, Dratman, Paul August, Gauge, EmilJ, Msh210, Oleg Alexandrov, Linas, Salix alba, HappyCamper, R.e.b., Goudzovski, YurikBot, Dmharvey, RussBot, Michael Slone, KSmrq, Gaius Cornelius, Gwaihir, Culli- nane, KnightRider~enwiki, Llanowan, Mhss, Bluebot, Silly rabbit, Nbarth, Harryboyles, Jim.belk, Zero sharp, Dycedarg, RobHar, Albmont, Spvo, Sullivan.t.j, Franp9am, Ixionid, Jeepday, Policron, Pleasantville, Drschawrz, YohanN7, YonaBot, JackSchmidt, An- chor Link Bot, ClueBot, Watchduck, Addbot, Roentgenium111, Topology Expert, Luckas-bot, Yobot, Ht686rg90, Ptbotgourou, Niout, Kilom691, AnomieBOT, Marconet, MondalorBot, Trappist the monk, Greenfernglade, John of Reading, ZéroBot, AvicAWB, Quondum, Emc2fred83, ClueBot NG, Brad7777, Moritorium, Mogism, Spectral sequence, Mark viking, CsDix, Diademodon, Some1Redirects4You and Anonymous: 35 • Glide reflection Source: https://en.wikipedia.org/wiki/Glide_reflection?oldid=679497811 Contributors: Patrick, Michael Hardy, Glenn, Charles Matthews, Henrygb, Tosha, Tomruen, Wgw4, [email protected], Aholtman, Mathbot, CiaPan, Roboto de Ajvol, Smack- 42.15. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 299

Bot, Smith609, Ezrakilty, Cydebot, Thijs!bot, .anacondabot, David Eppstein, Katalaveno, Aboluay, ClueBot, Addbot, Ronhjones, Leszek Jańczuk, Luckas-bot, AnomieBOT, MathsPoetry, N4m3, Akerans, Solomon7968, Trevayne08, Brad7777, Kelvinsong, Dexbot and Anonymous: 7 • Homography Source: https://en.wikipedia.org/wiki/Homography?oldid=675631192 Contributors: Rodrigob, Michael Hardy, Starfarmer, AugPi, Gandalf61, BenFrantzDale, Rgdboer, Uncle G, BorgHunter, Rjwilmsi, AndyKali, Dougthebug, Krishnavedala, Adoniscik, Wave- length, Rich8rd, Trovatore, Ligand, SmackBot, KYN, Nbarth, Kostmo, Hongooi, Nixeagle, Squamate, Jim.belk, A. Parrot, Myasuda, Cydebot, Bobke, Jiuguang Wang, Singularitarian, Llorenzi, Lingwitt, MacBishop, Aaport, Synthebot, Paolo.dL, Lourakis, Scientist47, ClueBot, 7&6=thirteen, Savo msu, Addbot, OrlinKolev, Luckas-bot, Yobot, JackPotte, KamikazeBot, AnomieBOT, Onthewings, Citation bot, Xaneris, PigFlu Oink, DrilBot, Ebony Jackson, Declan Clam, Ambrevar, Dinamik-bot, RjwilmsiBot, John of Reading, Slawekb, Jon- patterns, Quondum, D.Lazard, Wcherowi, BG19bot, Lxlxlx82, Brad7777, Krzysztof.mataj, Mogism, Jochen Burghardt, Montananevada- girl, Francesconazzaro and Anonymous: 37 • Homothetic transformation Source: https://en.wikipedia.org/wiki/Homothetic_transformation?oldid=666140727 Contributors: Hep- haestos, Patrick, Michael Hardy, Charles Matthews, Johnleemk, Josh Cherry, Altenmann, MathMartin, Pascalromon, Dbenbenn, DanielZM, Zaslav, Rgdboer, EmilJ, Olivier Mengué, Pontus, Oleg Alexandrov, Uncle G, Mpatel, Smmurphy, Mike Segal, FlaBot, Mathbot, Siddhant, YurikBot, Splintercellguy, Dbfirs, Mistercow, Cjfsyntropy, Minnesota1, SmackBot, Nbarth, Kostmo, Radagast83, CmdrObot, MaxEnt, Cydebot, Dlegland, Thijs!bot, Salgueiro~enwiki, Alphachimpbot, David Eppstein, Lantonov, Buckzuz, VolkovBot, BlueCanary9999, Ikarib, SieBot, Portalian, BotMultichill, Paolo.dL, Aboluay, Rinconsoleao, Marino-slo, Mild Bill Hiccup, Alexbot, PixelBot, Marc van Leeuwen, Addbot, LaaknorBot, Luckas-bot, NickK, MauritsBot, Ssola, RibotBOT, FrescoBot, Sławomir Biały, Tcnuk, ToematoeAdmn, BertSeghers, Anita5192, DjJelt, MusikAnimal, Blink11, Brad7777, Gastonlag, Undersum, BethNaught, Loraof and Anonymous: 25 • Improper rotation Source: https://en.wikipedia.org/wiki/Improper_rotation?oldid=679498170 Contributors: AxelBoldt, Zundark, The Anome, Patrick, Michael Hardy, TakuyaMurata, Stevenj, Charles Matthews, Hyacinth, Rogper~enwiki, Seanohagan, Giftlite, Dratman, JeffBobFrank, Tomruen, Bender235, Eric Kvaalen, Oleg Alexandrov, KSmrq, SmackBot, Mhss, WhiteHatLurker, Usgnus, Mattisse, Thijs!bot, David Eppstein, Lantonov, VolkovBot, Paolo.dL, Addbot, AndersBot, MathsPoetry, BenzolBot, Quondum, JellyPatotie and Anonymous: 4 • Inverse function Source: https://en.wikipedia.org/wiki/Inverse_function?oldid=682224507 Contributors: AxelBoldt, Tarquin, Bdesham, Michael Hardy, Looxix~enwiki, Glenn, Poor Yorick, Charles Matthews, Dcoetzee, Dysprosia, Jitse Niesen, Greenrd, Saltine, Kwantus, Mina86, Gromlakh, Robbot, Fredrik, Scarfboy, Bkell, Tobias Bergemann, Tosha, Giftlite, Qartis, John Palkovic, Icairns, PhotoBox, Discospinster, Guanabot, ObsessiveMathsFreak, ReiVaX, Paul August, Kenb215, El C, Passw0rd, EvenT, Infobacker, Oleg Alexandrov, Woohookitty, MFH, Ryan Reich, Graham87, Qwertyus, Rjwilmsi, JVz, Salix alba, VKokielov, ChongDae, CiaPan, Chobot, Krish- navedala, YurikBot, Wavelength, KSmrq, Rick Norwood, Bota47, Kompik, Gesslein, Banus, SmackBot, Incnis Mrsi, Eskimbot, Xaos- flux, Nbarth, SundarBot, Wen D House, Jon Awbrey, Lambiam, Jim.belk, Dr Greg, Hvn0413, Dreftymac, Happy-melon, JRSpriggs, Arabic Pilot, Runningonbrains, Mct mht, Gregbard, Sam Staton, Blindman shady, Thijs!bot, Kilva, Lt. CiberShark~enwiki, Paquitotrek, EdJohnston, Escarbot, TK-925, Béka, Gcm, Unifey~enwiki, Burga, VoABot II, Tibordp, Planemo, Jwuthe2, Haseldon, Policron, Lo- kiClock, PMajer, Anonymous Dissident, Clark Kimberling, Broadbot, Synthebot, AlleborgoBot, EverGreg, EmxBot, Ken Kuniyuki, Quietbritishjim, SieBot, Ivan Štambuk, RJaguar3, Paolo.dL, Harry~enwiki, Randomblue, Wahrmund, ClueBot, Rustic, Marino-slo, The Thing That Should Not Be, Ldimicco, Mild Bill Hiccup, Excirial, Zlewis101, Addbot, Download, Ehrenkater, Lightbot, Jarble, Legobot, Luckas-bot, Yobot, Fraggle81, Legobot II, KamikazeBot, AnomieBOT, JackieBot, Farhil, Hombre1729, Flewis, E2eamon, Diego Queiroz, DannyAsher, Xqbot, Sionus, NOrbeck, SassoBot, ViolaPlayer, FrescoBot, OgreBot, 00Ragora00, I dream of horses, Belovedeagle, Suffusion of Yellow, Hrvatistan, DASHBot, EmausBot, RA0808, Wikipelli, RaptureBot, Anti-min, Chewings72, ClueBot NG, Wcherowi, Matthiaspaul, Escapepea, Helpful Pixie Bot, Leonxlin, MadamIamadam, Frze, Matha288, JayEB, Brad7777, Mogism, PC-XT, Sogenius, DavidLeighEllis, Werddemer, Danielsevero, Wikiyroc, Kkim10, Fmmmlee, Philologick and Anonymous: 163 • Linear function (calculus) Source: https://en.wikipedia.org/wiki/Linear_function_(calculus)?oldid=625067690 Contributors: Gandalf61, Incnis Mrsi, CBM, Andyjsmith, Magioladitis, Yobot, Materialscientist, Fox Wilson, D.Lazard, Staszek Lem, Wcherowi, Lfahlberg, Babitaarora, Citrusbowler and Anonymous: 3 • Linear map Source: https://en.wikipedia.org/wiki/Linear_map?oldid=685925890 Contributors: AxelBoldt, Zundark, Tarquin, Taw, Toby Bartels, Miguel~enwiki, Stevertigo, Patrick, Chas zzz brown, Michael Hardy, Wshun, Glenn, Andres, Charles Matthews, Dysprosia, Jitse Niesen, Timc, Maximus Rex, Fibonacci, Moriel~enwiki, Robbot, Jaredwf, Giftlite, Inkling, Lethe, Lupin, Fropuff, Wwoods, Ra- dius, Sreyan, Noe, Lockeownzj00, Profvk, Maximaximax, RainerBlome, Jbinder, Barnaby dawson, ArnoldReinhold, Paul August, Zaslav, Rgdboer, Raazer, Obradovic Goran, HasharBot~enwiki, Jumbuck, Crispiness, Oleg Alexandrov, Joriki, Mosteo, Camw, Mekong Blues- man, Graham87, Thoughtactivist, Seidenstud, Rvmiller89, Salix alba, Juan Marquez, FlaBot, VKokielov, Jrtayloriv, Chobot, CAD6DEE2E8DAD95A, Zingus, B-Con, Vanished user 1029384756, E2mb0t~enwiki, Crasshopper, BOT-Superzerocool, Bota47, Pred, Bo Jacoby, Sbyrnes321, SmackBot, Mmernex, InverseHypercube, KocjoBot~enwiki, Jcarroll, Mhss, Kmarinas86, Bh3u4m, Nbarth, Kostmo, DHN-bot~enwiki, Truelight, Tbjw, Richard L. Peterson, Deditos, AB, Jim.belk, Dicklyon, Mets501, Hetar, Brandon rioja, Sinhautkarsh, JRSpriggs, Anakata, Cydebot, Ntsimp, Juansempere, Rlupsa, Edchi, Futurebird, JAnDbot, Ptery, Albmont, Jakob.scholbach, Soulbot, JJ Harrison, Sullivan.t.j, AsgardBot, David Eppstein, Numbo3, Wayp123, Lantonov, Tarotcards, Haseldon, Ross Fraser, LokiClock, Geometry guy, RiverStyx23, Eubulides, PedroV100, Wolfrock, Quietbritishjim, Paolo.dL, Thehotelambush, JackSchmidt, Fox816, Iandiver, Ldimicco, JP.Martin- Flatin, Erudecorp, He7d3r, Bender2k14, Farisori, Horizonwards, Kaedenn, Marc van Leeuwen, SilvonenBot, Galzigler, Addbot, Roent- genium111, IOLJeff, TeH nOmInAtOr, Jarble, Legobot, Luckas-bot, Yobot, TaBOT-zerem, Legobot II, AnomieBOT, ThinkerFeeler, Götz, Bdmy, Isheden, Niofis, Thehelpfulbot, Anterior1, Sławomir Biały, Arctic Night, Stpasha, Red Denim, Jcap0521, WildBot, Nø, Netheril96, Quondum, Vilietha, Jalexander-WMF, ChuispastonBot, Anita5192, Ray80127, Kasirbot, Aea6y9, Brad7777, Miguelcruzf, BattyBot, Ovikholt, Ariaveeg, Jpeterson1346 and Anonymous: 104 • Matrix (mathematics) Source: https://en.wikipedia.org/wiki/Matrix_(mathematics)?oldid=687602964 Contributors: AxelBoldt, Tar- quin, Tbackstr, Hajhouse, XJaM, Ramin Nakisa, Stevertigo, Patrick, Michael Hardy, Wshun, Cole Kitchen, SGBailey, Chinju, Zeno Gantner, Dcljr, Ejrh, Looxix~enwiki, Muriel Gottrop~enwiki, Angela, Александър, Poor Yorick, Rmilson, Andres, Schneelocke, Charles Matthews, Dysprosia, Jitse Niesen, Lou Sander, Dtgm, Bevo, J D, Francs2000, Robbot, Mazin07, Sander123, Chrism, Fredrik, R3m0t, Gandalf61, MathMartin, Sverdrup, Rasmus Faber, Bkell, Paul Murray, Neckro, Tobias Bergemann, Tosha, Giftlite, Jao, Arved, Ben- FrantzDale, Netoholic, Herbee, Dissident, Dratman, Michael Devore, Waltpohl, Duncharris, Macrakis, Utcursch, Alexf, MarkSweep, Profvk, Wiml, Urhixidur, Sam nead, Azuredu, Barnaby dawson, Porges, PhotoBox, Shahab, Rich Farmbrough, FiP, ArnoldReinhold, Pavel Vozenilek, Paul August, ZeroOne, El C, Rgdboer, JRM, NetBot, The strategy freak, La goutte de pluie, Obradovic Goran, Mdd, Tsirel, LutzL, Landroni, Jumbuck, Jigen III, Alansohn, ABCD, Fritzpoll, Wanderingstan, Mlm42, Jheald, Simone, RJFJR, Dirac1933, AN(Ger), Adrian.benko, Oleg Alexandrov, Nessalc, Woohookitty, Igny, LOL, Webdinger, David Haslam, UbiquitousUK, Username314, 300 CHAPTER 42. VECTOR SPACE

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Recorder, ChrisMiddleton, GirasoleDE, Do- gah, SieBot, Ivan Štambuk, Bachcell, Gerakibot, Cwkmail, Yintan, Radon210, Elcobbola, Paolo.dL, Oxymoron83, Ddxc, Oculi, Manway, AlanUS, Anchor Link Bot, Rinconsoleao, Denisarona, Canglesea, Myrvin, DEMcAdams, ClueBot, Sural, Wpoely86, Remag Kee, Super- Hamster, LizardJr8, Masterpiece2000, Excirial, Da rulz07, Bender2k14, Ftbhrygvn, Muhandes, Brews ohare, Tyler, Livius3, Jotterbot, Hans Adler, Manco Capac, MiraiWarren, Qwfp, Johnuniq, TimothyRias, Lakeworks, XLinkBot, Marc van Leeuwen, Rror, AndreNatas, Jaan Vajakas, Porphyro, Stephen Poppitt, Addbot, Proofreader77, Deepmath, RPHv, Steve.jaramillov~enwiki, WardenWalk, Jccwiki, CactusWriter, Mohamed Magdy, MrOllie, Tide rolls, Gail, Jarble, CountryBot, LuK3, Luckas-bot, Yobot, Senator Palpatine, Queen- Cake, TestEditBot, AnomieBOT, Autarkaw, Gazzawi, Archon 2488, IDangerMouse, MattTait, Kingpin13, Materialscientist, Citation bot, Wrelwser43, LilHelpa, FactSpewer, Xqbot, Capricorn42, Drilnoth, HHahn, El Caro, BrainFRZ, J04n, Nickmn, RibotBOT, Cerni- agigante, Smallman12q, WaysToEscape, Much noise, LucienBOT, Tobby72, VS6507, Recognizance, Sławomir Biały, Izzedine, IT2000, HJ Mitchell, Sae1962, Jamesooders, Cafreen, Citation bot 1, Swordsmankirby, I dream of horses, Kiefer.Wolfowitz, MarcelB612, NoFly- ingCars, RedBot, RobinK, Kallikanzarid, Jordgette, ItsZippy, Vairoj, SeoMac, MathInclined, The last username left was taken, Birat lamichhane, Katovatzschyn, Soupjvc, Sfbaldbear, Salvio giuliano, Mandolinface, EmausBot, Lkh2099, Nurath224, DesmondSteppe, RIS cody, Slawekb, Quondum, Chocochipmuffin, U+003F, Rcorcs, තඹරු විජේසේකර, Maschen, Babababoshka, Adjointh, Donner60, Puf- fin, JFB80, Anita5192, Petrb, ClueBot NG, Wcherowi, Michael P. Barnett, Rtucker913, Satellizer, Rank Penguin, Tyrantbrian, Dsperlich, Helpful Pixie Bot, Rxnt, Christian Matt, MarcoPotok, BG19bot, Wiki13, Muscularmussel, MusikAnimal, Brad7777, René Vápeník, Sofia karampataki, BattyBot, Freesodas, IkamusumeFan, Lucaspentzlp, OwenGage, APerson, Dexbot, Mark L MacDonald, Numbermaniac, Frosty, JustAMuggle, Reatlas, Acetotyce, Debouch, Wamiq, Ugog Nizdast, Zenibus, SwimmerOfAwesome, Jianhui67, OrthogonalFrog, Airwoz, Derpghvdyj, Mezafo, Botha42, CarnivorousBunny, Xxhihi, Sordin, Username89911998, Gronk Oz, Hidrolandense, Kellywacko, Frost.derec, Norbornene, Solid Frog, JArnold99, Kavya l, Mikeloud and Anonymous: 631 • Origin (mathematics) Source: https://en.wikipedia.org/wiki/Origin_(mathematics)?oldid=686735103 Contributors: Andre Engels, Sil- verfish, Charles Matthews, Bearcat, Giftlite, Taka, Gsutter, Jeremykemp, Discospinster, Rich Farmbrough, Wikiacc, Linas, OneWeird- Dude, The wub, Chobot, Whosasking, YurikBot, Silly rabbit, Octahedron80, Bjankuloski06en~enwiki, CBM, Equendil, Thijs!bot, Kon- radek, Braindrain0000, Penubag, VoABot II, David Eppstein, Infovarius, TXiKiBoT, Synthebot, SieBot, EliBerger, Paolo.dL, Beeble- brox, ClueBot, Marino-slo, Avoided, Addbot, AkhtaBot, SpBot, Zorrobot, Xqbot, JimVC3, Erik9bot, Pinethicket, Rausch, Dinamik-bot, MrX, Igor Yalovecky, EmausBot, John of Reading, Coasterlover1994, Donner60, Booradleyp, ClueBot NG, Widr, Vkpd11, Qetuth, Heskey.3387, Muhamad ittal and Anonymous: 33 • Orthogonal matrix Source: https://en.wikipedia.org/wiki/Orthogonal_matrix?oldid=678471608 Contributors: AxelBoldt, Tarquin, Patrick, Michael Hardy, Tim Starling, TakuyaMurata, Stevenj, Zhaoway~enwiki, Charles Matthews, Jitse Niesen, Robbot, 1984, Kaol, Math- Martin, Tosha, Giftlite, Fropuff, Dratman, Macrakis, Chris Howard, Paul August, Rgdboer, Echuck215, PAR, Oleg Alexandrov, LOL, BD2412, HappyCamper, Mathbot, Kcarnold, YurikBot, Wavelength, Vecter, KSmrq, Gaius Cornelius, NawlinWiki, Crasshopper, Light current, Danielx, Pred, SmackBot, Haymaker, BiT, Moocowpong1, Oli Filth, Silly rabbit, Kostmo, Rludlow, SundarBot, Jim.belk, Nij- dam, Yoderj, Nialsh, Ariel Pontes, Krasnoludek, Tawkerbot2, Myasuda, Mct mht, Countchoc, OrenBochman, JEBrown87544, Ben pcc, Salgueiro~enwiki, Erxnmedia, Coffee2theorems, Jakob.scholbach, David Eppstein, User A1, Pkrecker, JoergenB, Tercer, Shinigami Josh, Kesal, Policron, Burkhard.Plache, Kyap, Ezzaldeen, JohnBlackburne, TXiKiBoT, Vladsinger, Simogasp, AlleborgoBot, YonaBot, Da Joe, Paolo.dL, Blacklemon67, Rinconsoleao, Alexbot, Bender2k14, SchreiberBike, Humanengr, XLinkBot, Addbot, Roentgenium111, DOI bot, EconoPhysicist, Legobot, Luckas-bot, Yobot, Calle, 9258fahsflkh917fas, Citation bot, LilHelpa, The suffocated, LucienBOT, Grinevitski, Citation bot 1, NA Correct, Tkuvho, Tal physdancer, Hill2690, Bluefist, Diannaa, Alph Bot, WikitanvirBot, Quondum, Zueig- nung, ChrisGualtieri, Illia Connell, Latrace, Toussapace, Tzvy, Anas satti404, Kshithappens, Kfitzell29, JOEBLOGGES and Anonymous: 72 • Outline of calculus Source: https://en.wikipedia.org/wiki/Outline_of_calculus?oldid=635867478 Contributors: Charles Matthews, Gan- dalf61, Michael Devore, Maurreen, Mdd, Woohookitty, Quiddity, Googl, Pegship, Auroranorth, David Kernow, Nbarth, Onorem, Cy- bercobra, Nexus Seven, CBM, Xantharius, The Transhumanist, The Transhumanist (AWB), Geometry guy, Auntof6, Robert Skyhawk, Sneakysneaky29, Ozob, Estudiarme, AnomieBOT, Neurolysis, Charvest, Thehelpfulbot, FrescoBot, Tkuvho, Gamewizard71, Nuke- ofEarl, Ebirtog, Lugia2453, Dreamermaksud and Anonymous: 8 • Parallelogram Source: https://en.wikipedia.org/wiki/Parallelogram?oldid=687295572 Contributors: AxelBoldt, Youssefsan, XJaM, Michael Hardy, GABaker, Ixfd64, Eric119, Ellywa, Iammaxus, Julesd, Andres, Charles Matthews, Dysprosia, Gutza, Robbot, RedWolf, Academic Challenger, Intangir, Giftlite, Zuytdorp Survivor, MSGJ, Dratman, David Johnson, Eequor, Utcursch, Mamizou, Tomruen, Almit39, GNU, Trevor MacInnis, Grunt, Clubjuggle, Mariko~enwiki, Freakofnurture, EugeneZelenko, Discospinster, Cacycle, Aecis, Smalljim, Dungodung, Minghong, Storm Rider, Alansohn, Jamyskis, Cburnett, Mikeo, Agutie, Oleg Alexandrov, Linas, CS42, Kelisi, Noet- ica, Mandarax, Kbdank71, Sjakkalle, OneWeirdDude, Hiberniantears, SMC, Dracontes, FlaBot, AdnanSa, Mathbot, Gurch, Imnot- minkus, YurikBot, Adam1213, Phantomsteve, DigitalRosh, Yyy, Wimt, Grafen, SivaKumar, Anetode, KovacsUr, Dbfirs, Bota47, Rfs- mit, Fang Aili, Livitup, GraemeL, Paul D. Anderson, Paul Erik, SmackBot, Hydrogen Iodide, McGeddon, Jab843, Dohermike, Gilliam, Skizzik, Jeekc, MalafayaBot, Octahedron80, DHN-bot~enwiki, Tamfang, Ioscius, Vanished User 0001, TKD, Rsm99833, Addshore, 42.15. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 301

Whpq, SashatoBot, IronGargoyle, Stwalkerster, Optakeover, Limaner, Gholam, Lenoxus, Tawkerbot2, JForget, Tanthalas39, CBM, Tschel, MarsRover, A876, Perfect Proposal, Bazzargh, Raoul NK, JamesAM, Thijs!bot, Clarencewong, Epbr123, Sagaciousuk, Eric- qiu, Marek69, John254, WillMak050389, NorwegianBlue, James086, Mentifisto, John.d.page, Dr who1975, Bluetooth954, Tim Shuba, Braindrain0000, JAnDbot, Davewho2, MER-C, Andonic, PhilKnight, Magioladitis, VoABot II, AuburnPilot, JamesBWatson, Catgut, David Eppstein, Ptrpro, MartinBot, RisenAge, J.delanoy, Bogey97, Eliz81, LordAnubisBOT, Larion, Trasdahl~enwiki, Squids and Chips, VolkovBot, Jorioux, Indubitably, AlnoktaBOT, Philip Trueman, TXiKiBoT, Goldmember24, A4bot, Anonymous Dissident, Someguy1221, Martin451, JhsBot, Rjm at sleepers, Madhero88, Belamp, Jason Leach, Falcon8765, Artrush, Praefectorian, SieBot, Jsc83, 1234567890abz, Keilana, Breawycker, Happysailor, Flyer22 Reborn, Le Pied-bot~enwiki, JSpung, Sbowers3, Oxymoron83, Kudret abi, C'est moi, ClueBot, Runekolovell, Zero over zero, Meekywiki, Mild Bill Hiccup, CounterVandalismBot, ZerglingChamp, DragonBot, Quaffed, Aitias, Versus22, Stickee, Gerhardvalentin, WikiDao, HexaChord, D.M. from Ukraine, Addbot, Fgnievinski, Mr. Wheely Guy, Vishnava, CanadianLinuxUser, MrOllie, Download, LaaknorBot, PranksterTurtle, Kyle1278, Numbo3-bot, Tide rolls, PV=nRT, QuadrivialMind, MuZemike, Legobot, Luckas-bot, Yobot, Daren1997, Ptbotgourou, Fraggle81, Amirobot, Maxí, KamikazeBot, Demo- craticLuntz, Jim1138, Piano non troppo, AdjustShift, Sz-iwbot, Abshirdheere, Bluerasberry, Materialscientist, Lollenee, Xqbot, Capri- corn42, GrouchoBot, King Bong, Youmils03, Shadowjams, Aaron Kauppi, Erik9, Nenya17, FrescoBot, Pepper, Zarrgoth, StaticVision, Randolph3, FrankMJohnson, JackOL31, Pinethicket, I dream of horses, Timothychiam, Triplestop, Jschnur, SpaceFlight89, Contin- ueWithCaution, Meaghan, White Shadows, TobeBot, Ekanil, Uranus124, MrX, 777sms, Duoduoduo, Reaper Eternal, Seahorseruler, DARTH SIDIOUS 2, Mean as custard, Cdolan03, Slon02, EmausBot, Ikerus, Racerx11, DotKuro, Tommy2010, Winner 42, Wikipelli, K6ka, A930913, Nikhiljha, Mayur, Donner60, ChuispastonBot, ClueBot NG, Jack Greenmaven, Wcherowi, Lizzy2010, Vacation9, Ytrewq17, Widr, Titodutta, Jmmahony, ISTB351, MusikAnimal, AwamerT, Circlesareround, Spandaize, Brad7777, Klilidiplomus, Darylgolden, Calebbbbb, Aliwal2012, CarrieVS, LukeLAnderson, Lugia2453, Frosty, Sriharsh1234, Acathine, Rawrzar890, Ekips39, Altered Walter, Quae4511, AnthonyJ Lock, Rudy Cliff, Oblong1, SJ Defender, Amruth555, JaconaFrere, Vieque, Joehynes2257, Teach- erschool01, TerryAlex, GeoffreyT2000, Loraof, Drmurali13580, Devan1217 and Anonymous: 532 • Position (vector) Source: https://en.wikipedia.org/wiki/Position_(vector)?oldid=685831865 Contributors: Patrick, AugPi, Finlay McWal- ter, Giftlite, BenFrantzDale, Discospinster, Rgdboer, Gary, Diego Moya, Dzordzm, RuM, Thecurran, Chobot, Albedo, Cmglee, Chris the speller, MalafayaBot, A. B., Psykocyber, Ohconfucius, SashatoBot, Jambaugh, Benplowman, FilipeS, Cydebot, A876, Sommacal alfonso, ManN, JAnDbot, MarcusMaximus, JCraw, R'n'B, JohnBlackburne, Thurth, TXiKiBoT, Anonymous Dissident, Synthebot, Neparis, Do- gah, Gerakibot, Paolo.dL, OKBot, Mild Bill Hiccup, UKoch, Alexbot, Addbot, Fgnievinski, Mpfiz, Götz, Materialscientist, ArthurBot, Xqbot, Drilnoth, XZeroBot, FrescoBot, Dger, DrilBot, Htmlvb, Surya Adinata, EmausBot, Ready, Maschen, ClueBot NG, AeroPsico, Helpful Pixie Bot, Brad7777, Qetuth, BattyBot, Pbierre, Hghyux, Eyesnore, Muhamad ittal and Anonymous: 33 • Reflection (mathematics) Source: https://en.wikipedia.org/wiki/Reflection_(mathematics)?oldid=686864611 Contributors: The Anome, SimonP, Patrick, Mdupont, TakuyaMurata, Glenn, AugPi, Charles Matthews, Dysprosia, Phys, MathMartin, Tosha, Giftlite, Frop- uff, Tomruen, PhotoBox, Rich Farmbrough, Paul August, Gonzalo Diethelm, Zaslav, BenjBot, Rgdboer, Jet57, MiguelTremblay, Oleg Alexandrov, Joe Decker, 25~enwiki, Nneonneo, Yamamoto Ichiro, Eubot, [email protected], Mathbot, Gurch, Scythe33, Chobot, YurikBot, Wavelength, Shell Kinney, Gwaihir, Grafen, Phgao, MathsIsFun, Mhss, Tamfang, Mecrazywong, Goodnightmush, Jim.belk, Cydebot, Thijs!bot, Headbomb, Oemb1905, Salgueiro~enwiki, .anacondabot, David Eppstein, J.delanoy, Wandering Ghost, Comet- styles, RJASE1, Pleasantville, JohnBlackburne, LokiClock, TXiKiBoT, Ceranthor, Rknasc, Cryonic07, SieBot, Jsc83, Paolo.dL, DanDs, Aboluay, ClueBot, GorillaWarfare, Marino-slo, Jan1nad, Razorflame, Marc van Leeuwen, Addbot, Fgnievinski, Charcole125, Laaknor- Bot, Dayewalker, Luckas-bot, JackieBot, Materialscientist, Loveless, Dako1, Sławomir Biały, Tkuvho, DASHBot, EmausBot, Acather96, ScottyBerg, Wikipelli, K6ka, Slawekb, Quondum, ClueBot NG, Wcherowi, Brad7777, Fiboman, CopperSolder208, WillemienH, Graboy and Anonymous: 59 • Rigid body Source: https://en.wikipedia.org/wiki/Rigid_body?oldid=683567480 Contributors: XJaM, Peterlin~enwiki, Patrick, An- dres, Charles Matthews, Robbot, Giftlite, Fropuff, Starsong, Abdull, Discospinster, JimR, ZeroOne, Laurascudder, Jalexiou, Nk, Oleg Alexandrov, Linas, Armando, Melesse, Rjwilmsi, Mathbot, Chobot, Sanpaz, Algebraist, YurikBot, Borgx, Albedo, Bota47, Obakeneko, KocjoBot~enwiki, Bidgee, Silly rabbit, DHN-bot~enwiki, DMacks, Vina-iwbot~enwiki, Aboeing, Origin415, CmdrObot, Equendil, An- drewHowse, Cydebot, Simon Brady, Juansempere, AndrewDressel, Mojo Hand, JAnDbot, Cardamon, Shim'on, MarcusMaximus, M samadi, Iain marcuson, P.wormer, XCelam, JohnBlackburne, Thurth, SieBot, YonaBot, Da Joe, DaBler, Paolo.dL, JerroldPease-Atlanta, ClueBot, Ideal gas equation, The Thing That Should Not Be, Abdullah Köroğlu~enwiki, Brews ohare, Addbot, MrOllie, Yobot, E mraedarab, J04n, GrouchoBot, RibotBOT, , FrescoBot, Dger, Steve Quinn, Hoo man, Jauhienij, Cnwilliams, Jordgette, Tbhotch, EmausBot, ZéroBot, L Kensington, Svetsveti, ClueBot NG, CocuBot, Helpful Pixie Bot, BG19bot, Vagobot, Neøn, Jamontaldi, Lorien- drew, Am.sreenath, Monkbot, Ortiz80lobos and Anonymous: 43 • Rotation Source: https://en.wikipedia.org/wiki/Rotation?oldid=684475672 Contributors: The Anome, Tarquin, AstroNomer~enwiki, -- April, Ed Poor, Youssefsan, Peterlin~enwiki, Heron, Stevertigo, Patrick, Michael Hardy, TakuyaMurata, Gbleem, Kosebamse, Glenn, AugPi, Andres, Smack, Reddi, Dysprosia, Hyacinth, Robbot, Jaredwf, Fredrik, Altenmann, Nurg, Rholton, UtherSRG, Cutler, Adam78, Tosha, Giftlite, Dbenbenn, DocWatson42, Harp, Mintleaf~enwiki, Wolfkeeper, BenFrantzDale, MSGJ, Micru, Andycjp, Antandrus, Karol Langner, Oneiros, MFNickster, JimWae, Phil1988, CALR, Dcfleck, ZeroOne, RJHall, El C, Bobo192, I9Q79oL78KiL0QTFHgyc, Ranveig, Katefan0, Snowolf, Jheald, David Prentiss, Recury, Oleg Alexandrov, Tom.k, Wedesoft, Jeff3000, Gerbrant, Graham87, Cuchul- lain, Jshadias, Jorunn, Rjwilmsi, MarSch, LjL, Mathbot, Ewlyahoocom, DVdm, Wingchi, Algebraist, Siddhant, YurikBot, Hairy Dude, Epolk, KSmrq, Friedfish, Gaius Cornelius, NawlinWiki, Semperf, Zwobot, CLW, Dna-webmaster, MathsIsFun, Incnis Mrsi, Eskim- bot, Dyersgoodness, Edgar181, Gaff, Yamaguchi, Saros136, Bluebot, GBE, Stephen.frede, Nbarth, Modest Genius, Suicidalhamster, Tsca.bot, SundarBot, Will Beback, Eliyak, BorisFromStockdale, NYCJosh, Peter Horn, Gnahz99, Levineps, Iridescent, UncleDoug- gie, Sjoerd22, MarsRover, WeggeBot, Equendil, AndrewHowse, Cydebot, Abeg92, Juansempere, Satori Son, Epbr123, Willworkforice- cream, Marek69, Kborer, AntiVandalBot, Widefox, Seaphoto, Res2216firestar, JAnDbot, ThomasO1989, MER-C, PhilKnight, VoABot II, MartinDK, Swpb, JMyrleFuller, InkKznr, Robin S, TheEgyptian, J.delanoy, M samadi, Pharaoh of the Wizards, AstroHurricane001, Ignatzmice, Hut 6.5, Fountains of Bryn Mawr, Policron, Inwind, CardinalDan, RJASE1, VolkovBot, Pleasantville, JohnBlackburne, Davidwr, Pamplelune, Martin451, Dirkbb, VanBuren, Schenectady, SieBot, Ttony21, Portalian, RJaguar3, Paolo.dL, OKBot, Anchor Link Bot, WikiLaurent, Denisarona, ClueBot, The Thing That Should Not Be, Tomcj, Jan1nad, Excirial, PhySusie, Bremerenator, Muro Bot, Riott Nyte, SoxBot III, DumZiBoT, Badgernet, On the other side, Addbot, Willking1979, Corvus Fox, Atethnekos, Fgnievinski, Sri- harsha.karnati, Dubito ergo sum, Numbo3-bot, Tide rolls, Romanskolduns, Legobot, Luckas-bot, Yobot, Konsty, Dhruv.mongia, Chuck- iesdad, Materialscientist, ArthurBot, Plastadity, Sionus, JimVC3, Capricorn42, Bihco, Magicxcian, J04n, Calculus HK, SylvieHorse, Erik9, Taychef, D'ohBot, Sławomir Biały, Parvons, Allendaves, PGNicolay, Pinethicket, HRoestBot, Tom.Reding, JasmineVioletWin- ston, Zbayz, Tim1357, Abc518, Jordgette, Lotje, Vrenator, DexDor, Hajatvrc, EmausBot, Orphan Wiki, R*elation, Jmencisom, JSquish, ZéroBot, Carlitos0620, Quondum, ImyourWikiDude, Isaac Euler, Donner60, Bonty007bond, ChuispastonBot, ClueBot NG, PoqVaUSA, 302 CHAPTER 42. VECTOR SPACE

Doh5678, Widr, FakTNeviM, Bryolvera, Anbu121, IkamusumeFan, Pbierre, Ben525, ChrisGualtieri, TheJJJunk, EuroCarGT, VicGuy, Lugia2453, Sethm123, ForcedLogix, Milf69ers, Mario Castelán Castro, Loraof, Palak46 and Anonymous: 217 • Scaling (geometry) Source: https://en.wikipedia.org/wiki/Scaling_(geometry)?oldid=681956779 Contributors: Patrick, Glenn, Charles Matthews, Jaredwf, Giftlite, Jorge Stolfi, Mike Rosoft, MichaelMcGuffin, Rich Farmbrough, VBGFscJUn3, BD2412, Martin von Gagern, Daniel Mietchen, Dbfirs, Mhss, Nbarth, Aboeing, Kelly elf, Cydebot, Thijs!bot, JaGa, Schmloof, Lantonov, VolkovBot, Pleasantville, Imasleepviking, Burntsauce, Paolo.dL, ClueBot, PipepBot, BOTarate, Qwfp, Little Mountain 5, Addbot, ArthurBot, Georg Stillfried, Omnipaedista, FrescoBot, Wikipelli, SporkBot, AManWithNoPlan, ClueBot NG, Faizanalivarya, Brad7777, Stephen Balaban, Loraof and Anonymous: 28 • Semidirect product Source: https://en.wikipedia.org/wiki/Semidirect_product?oldid=672544245 Contributors: AxelBoldt, Bryan Derk- sen, Zundark, Edward, Patrick, Chas zzz brown, Michael Hardy, TakuyaMurata, Charles Matthews, Phys, Jleedev, Tosha, Giftlite, Lethe, Lupin, MSGJ, Fropuff, Elroch, Zaslav, Gauge, Kundor, Firsfron, Tabletop, Grammarbot, Juan Marquez, R.e.b., FlaBot, Algebraist, Space- potato, KSmrq, Grubber, RonnieBrown, PeterKoroteev, SmackBot, Melchoir, Jjalexand, Nbarth, Vina-iwbot~enwiki, Aghitza, Jim.belk, Michael Kinyon, Tyrrell McAllister, Noleander, CmdrObot, TheTito, Mathisreallycool, Magioladitis, JamesBWatson, Jakob.scholbach, Cuzkatzimhut, VolkovBot, AlleborgoBot, YohanN7, JackSchmidt, He7d3r, ATC2, Algebran, Sabalka, Addbot, Мыша, Calculuslover, Legobot, TommasoT, AnomieBOT, Materialscientist, FrescoBot, CESSMASTER, Lotje, ZéroBot, Quondum, Tijfo098, Aeginn, Chris- Gualtieri, Jhemelae, Makecat-bot, CsDix, Zoydb2, Fabiangabel, Nicofreeride, Some1Redirects4You and Anonymous: 49 • Shear mapping Source: https://en.wikipedia.org/wiki/Shear_mapping?oldid=672138592 Contributors: Bryan Derksen, Michael Hardy, Glenn, Charles Matthews, Jeffq, Sverdrup, Jorge Stolfi, Rubik-wuerfel, Gauge, Rgdboer, Danhash, Jheald, Oleg Alexandrov, Feezo, Felix- damrau~enwiki, YurikBot, Mhss, Geologyguy, Cydebot, Konradek, RobHar, Geekdiva, Pleasantville, Wikiisawesome, Rudsky, Addbot, Bunnyhop11, AnomieBOT, LilHelpa, Jambo6c, Wikfr, Aughost, ClueBot NG, Brad7777, Jochen Burghardt, Tgbtg316 and Anonymous: 17 • Similarity (geometry) Source: https://en.wikipedia.org/wiki/Similarity_(geometry)?oldid=686473584 Contributors: AxelBoldt, Tar- quin, PierreAbbat, Patrick, Michael Hardy, Wshun, Kku, Dcljr, 6birc, AugPi, Charles Matthews, Dysprosia, Aleph4, Robbot, Bern- hard Bauer, Gandalf61, Tosha, Giftlite, Dbenbenn, BenFrantzDale, Tom harrison, Herbee, Running, Rama, Gauge, Rgdboer, RoyBoy, Bobo192, Johnkarp, Foobaz, Happenstantially~enwiki, Dirac1933, Agutie, Woohookitty, Splintax, Mpatel, Isnow, Noetica, Bubuka, HannsEwald, Salix alba, Mathbot, Margosbot~enwiki, Chobot, DVdm, YurikBot, Karlscherer3, Gene.arboit, RussBot, Bleakcomb, Zwobot, Dbfirs, Fram, Halcyonhazard, Tropylium, Ghazer~enwiki, SmackBot, Bluebot, Silly rabbit, Svein Olav Nyberg, Henning Makholm, John Reid, Spiritia, Lambiam, Expedition brilliancy, IronGargoyle, RomanSpa, WhiteHatLurker, Iridescent, Geekygator, Erzbischof, Krauss, Dynaflow, Thijs!bot, Porqin, Snbritishfreak, Mhaitham.shammaa, Salgueiro~enwiki, Justinhwang1996, Freshacconci, Magiola- ditis, Seberle, David Eppstein, Calltech, Kayau, Keith D, PrestonH, J.delanoy, Katalaveno, Isall, Don4of4, Slysplace, Dmcq, Allebor- goBot, SieBot, Paolo.dL, Jonlandrum, Dolphin51, ClueBot, Marino-slo, Mild Bill Hiccup, Vrkunkel, DragonBot, BOTarate, Vegetator, XLinkBot, Tom.schramm~enwiki, SilvonenBot, NellieBly, Addbot, ConCompS, MrOllie, Download, Zorrobot, Luckas-bot, Amirobot, AnomieBOT, Jim1138, Materialscientist, ArthurBot, LilHelpa, Erud, DSisyphBot, Mr.gondolier, Armbrust, Prunesqualer, Youmils03, FrescoBot, Se0808, PigFlu Oink, Pinethicket, I dream of horses, Salvidrim!, Nassr, TobeBot, Mr.Prithz, Duoduoduo, Alph Bot, Nerdy- ScienceDude, EmausBot, UsüF, Susfele, Hazard-SJ, Maschen, ClueBot NG, Serasuna, Wcherowi, Satellizer, Shinli256, FightingMac, ISTB351, AdityaRaj16, Comfr, Dysrhythmia, Hghyux, Khazar2, Jochen Burghardt, Pedro Listel, Lucindali, Arakk123, Jfung1999, Lo- raof and Anonymous: 105 • Squeeze mapping Source: https://en.wikipedia.org/wiki/Squeeze_mapping?oldid=685644767 Contributors: Michael Hardy, Glenn, Charles Matthews, Dysprosia, Giftlite, Gauge, Rgdboer, Vengeful Cynic, Jheald, Nbarth, YohanN7, Yobot, NOrbeck, FrescoBot, Thecheesykid, Maschen, Isocliff, BG19bot, ChrisGualtieri, Jochen Burghardt, Katterjohn and Anonymous: 4 • Transformation matrix Source: https://en.wikipedia.org/wiki/Transformation_matrix?oldid=681015566 Contributors: Tbackstr, Patrick, Charles Matthews, Andrewman327, Tpbradbury, Chuunen Baka, Sverdrup, Jleedev, Giftlite, BenFrantzDale, Radius, Jorge Stolfi, Lu- casVB, Lockeownzj00, Rgdboer, Sietse Snel, Longhair, Ashokcm, Oleg Alexandrov, Dysepsion, Reedbeta, FlaBot, Mathbot, King of Hearts, Michael Slone, Grafen, Gareth Jones, BOT-Superzerocool, MaxDZ8, Light current, Fsiler, Mebden, Cmglee, SmackBot, Lar- sPensjo~enwiki, InverseHypercube, Thorseth, DHN-bot~enwiki, Hongooi, Javalenok, Tsca.bot, Tamfang, Xagent86, Vina-iwbot~enwiki, Melody Concerto, Rosasco, Cydebot, MuTau, Ninjakannon, Steelpillow, Charibdis, Lihan161051, Edward321, KleinKlio~enwiki, Bo- Josley, Cezn, JohnBlackburne, LokiClock, Philip Trueman, JhsBot, Jackfork, Amog, Paolo.dL, Anchor Link Bot, ClueBot, Arakunem, ,Luckas-bot, Amirobot, Wonderfl, Alexkin, AnomieBOT ,خالد حسني ,Marcosagliano, Stickee, Addbot, Zulon, Hand Poet Open, Jarble Qorilla, Martnym, Sagrael, Sławomir Biały, Jadave234, Updatehelper, Ennustaja, ClueBot NG, Mesoderm, Ssimasanti, Brad7777, Ido66667, JohnMathTeacher, BattyBot, Mark L MacDonald, Hotwebmatter, Lsmll, Ginsuloft, Jpamills, Barceloco, Dkong7, Youming Chen and Anonymous: 170 • Translation (geometry) Source: https://en.wikipedia.org/wiki/Translation_(geometry)?oldid=683768025 Contributors: Arvindn, Toby Bartels, SimonP, Patrick, Michael Hardy, Looxix~enwiki, Docu, Glenn, Andres, Charles Matthews, Dysprosia, Robbot, Jaredwf, P0lyglut, Tobias Bergemann, Filemon, Tosha, Giftlite, Lockeownzj00, Sonett72, Running, Rich Farmbrough, ArnoldReinhold, Paul August, Mister- Sheik, Rgdboer, Kwamikagami, Bobo192, Longhair, TheSolomon, Alansohn, Burn, CloudNine, Oleg Alexandrov, MONGO, Marudub- shinki, Mandarax, Dpv, MikeJ9919, [email protected], Mathbot, RexNL, Kri, GangofOne, Siddhant, YurikBot, Michael Slone, MathsIsFun, SmackBot, Maksim-e~enwiki, Unyoyega, Eskimbot, Danikolt, Tsca.bot, SashatoBot, JoshuaZ, Aboeing, Cydebot, Pas- cal.Tesson, Riojajar~enwiki, Thijs!bot, Mojo Hand, Hmrox, AntiVandalBot, Magioladitis, VoABot II, David Eppstein, Infovarius, Ariel., NewEnglandYankee, VolkovBot, Pleasantville, SieBot, Portalian, BotMultichill, Revent, Jdoggchief, Paolo.dL, Aboluay, Anchor Link Bot, ClueBot, Cliff, Thehelpfulone, SoxBot III, Mjoachimiak, Addbot, Tide rolls, Luckas-bot, Calle, SwisterTwister, AnomieBOT, Ma- terialscientist, Noideta, GrouchoBot, T3htao, FrescoBot, Innotata, Jowa fan, EmausBot, Chaohuang, K6ka, Aughost, Kilopi, JoeSper- razza, Jay-Sebastos, Anita5192, ClueBot NG, Mesoderm, Helpful Pixie Bot, Prof McCarthy, Solomon7968, Brad7777, JYBot, JPaquim, XenoPheX and Anonymous: 55 • Vector space Source: https://en.wikipedia.org/wiki/Vector_space?oldid=686815834 Contributors: AxelBoldt, Bryan Derksen, Zundark, The Anome, Taw, Awaterl, Youandme, N8chz, Olivier, Tomo, Patrick, Michael Hardy, Tim Starling, Wshun, Nixdorf, Kku, Gabbe, Wapcaplet, TakuyaMurata, Pcb21, Iulianu, Glenn, Ciphergoth, Dysprosia, Jitse Niesen, Jogloran, Phys, Kwantus, Aenar, Robbot, Ro- manm, P0lyglut, Tobias Bergemann, Giftlite, BenFrantzDale, Lethe, MathKnight, Fropuff, Waltpohl, Andris, Daniel Brockman, Python eggs, Chowbok, Sreyan, Lockeownzj00, MarkSweep, Profvk, Maximaximax, Barnaby dawson, Mh, Klaas van Aarsen, TedPavlic, Rama, Smyth, Notinasnaid, Paul August, Bender235, Rgdboer, Shoujun, Army1987, Cmdrjameson, Stephen Bain, Tsirel, Msh210, Orimosen- zon, ChrisUK, Ncik~enwiki, Eric Kvaalen, ABCD, Sligocki, Jheald, Eddie Dealtry, Dirac1933, Woodstone, Kbolino, Oleg Alexandrov, 42.15. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 303

Woohookitty, Mindmatrix, ^demon, Hfarmer, Mpatel, MFH, Graham87, Ilya, Rjwilmsi, Koavf, MarSch, Omnieiunium, Salix alba, Titoxd, FlaBot, VKokielov, Therearenospoons, Nihiltres, Ssafarik, Srleffler, Kri, R160K, Chobot, Gwernol, Algebraist, YurikBot, Wave- length, Spacepotato, Hairy Dude, RussBot, Michael Slone, CambridgeBayWeather, Rick Norwood, Kinser, Guruparan, Trovatore, Van- ished user 1029384756, Nick, Bota47, BraneJ, Martinwilke1980, Antiduh, Arthur Rubin, Lonerville, Netrapt, Curpsbot-unicodify, Cjf- syntropy, Paul D. Anderson, GrinBot~enwiki, SmackBot, RDBury, InverseHypercube, KocjoBot~enwiki, Davidsiegel, Chris the speller, SlimJim, SMP, Silly rabbit, Complexica, Nbarth, DHN-bot~enwiki, Colonies Chris, Chlewbot, Vanished User 0001, Cícero, Cybercobra, Daqu, Mattpat, James084, Lambiam, Tbjw, Breno, Terry Bollinger, Michael Kinyon, Lim Wei Quan, Rcowlagi, SandyGeorgia, Whack- awhackawoo, Inquisitus, Rschwieb, Levineps, Madmath789, Markan~enwiki, Tawkerbot2, Igni, CRGreathouse, Mct mht, Cydebot, Dan- man3459, Guitardemon666, Mikewax, Thijs!bot, Headbomb, RobHar, CharlotteWebb, Urdutext, Escarbot, JAnDbot, Thenub314, En- glebert, Magioladitis, Jakob.scholbach, Kookas, SwiftBot, WhatamIdoing, David Eppstein, Cpl Syx, Charitwo, Akhil999in, Infovarius, Frenchef, TechnoFaye, CommonsDelinker, Paranomia, Michaelp7, Mitsuruaoyama, Trumpet marietta 45750, Daniele.tampieri, Gom- bang, Policron, Fylwind, Cartiod, Camrn86, AlnoktaBOT, Hakankösem~enwiki, TXiKiBoT, Hlevkin, Gwib, Anonymous Dissident, Imasleepviking, Hrrr, Mechakucha, Geometry guy, Terabyte06, Tommyinla, Wikithesource, Staka, AlleborgoBot, Deconstructhis, New- byguesses, YohanN7, SieBot, Ivan Štambuk, Portalian, ToePeu.bot, Lucasbfrbot, Tiptoety, Paolo.dL, Henry Delforn (old), Thehotelam- bush, JackSchmidt, Jorgen W, AlanUS, Randomblue, Jludwig, ClueBot, Alksentrs, Nsk92, JP.Martin-Flatin, FractalFusion, Niceguyedc, DifferCake, Auntof6, 0ladne, PixelBot, Brews ohare, Jotterbot, Hans Adler, SchreiberBike, Jasanas~enwiki, Humanengr, TimothyR- ias, BodhisattvaBot, SilvonenBot, Jaan Vajakas, Addbot, Gabriele ricci, AndrewHarvey4, Topology Expert, NjardarBot, Looie496, Un- cia, ChenzwBot, Ozob, Wikomidia, TeH nOmInAtOr, Jarble, CountryBot, Yobot, Kan8eDie, THEN WHO WAS PHONE?, Edurazee, AnomieBOT, ^musaz, Götz, Citation bot, Xqbot, Txebixev, GeometryGirl, Point-set topologist, RibotBOT, Charvest, Quartl, Lisp21, FrescoBot, Nageh, Rckrone, Sławomir Biały, Citation bot 1, Kiefer.Wolfowitz, Jonesey95, MarcelB612, Stpasha, Mathstudent3000, Ju- jutacular, Dashed, TobeBot, Javierito92, January, Setitup, TjBot, EmausBot, WikitanvirBot, Brydustin, Fly by Night, Slawekb, Chricho, Ldboer, Quondum, D.Lazard, Milad pourrahmani, RaptureBot, Cloudmichael, ClueBot NG, Wcherowi, Chitransh Gaurav, Jiri 1984, Joel B. Lewis, Widr, Helpful Pixie Bot, Ma snx, David815, Alesak23, Probability0001, JOSmithIII, Duxwing, PsiEpsilon, IkamusumeFan, Աննա Շահինյան, IPWAI, JYBot, Dexbot, Catclock, Tch3n93, Fycafterpro, CsDix, Hella.chillz, Jose Brox, François Robere, Logan- falco, Newestcastleman, UY Scuti, K9re11, Monkbot, AntiqueReader, Aditya8795, KurtHeckman, Isambard Kingdom, Shivakrishna .Srinivas. Dasari, NateLloydClark and Anonymous: 216

42.15.2 Images • File:0001_0001_0001_1110_nonlinearity.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/66/0001_0001_0001_1110_ nonlinearity.svg License: Public domain Contributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk) • File:2D_affine_transformation_matrix.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/2c/2D_affine_transformation_ matrix.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Cmglee • File:3D_Cartesian_Coodinate_Handedness.jpg Source: https://upload.wikimedia.org/wikipedia/commons/b/b2/3D_Cartesian_Coodinate_ Handedness.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: Primalshell • File:45,_-315,_and_405_co-terminal_angles.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/56/45%2C_-315%2C_ and_405_co-terminal_angles.svg License: Public domain Contributors: Own work Original artist: Adrignola • File:5-cell.gif Source: https://upload.wikimedia.org/wikipedia/commons/d/d8/5-cell.gif License: Public domain Contributors: Trans- ferred from en.wikipedia to Commons. Original artist: JasonHise at English Wikipedia • File:Absolute_value_composition.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/ba/Absolute_value_composition. svg License: CC0 Contributors: Own work Original artist: Incnis Mrsi • File:Academ_Example_of_similarity_with_ratio_square_root_of_2.svg Source: https://upload.wikimedia.org/wikipedia/commons/ 3/33/Academ_Example_of_similarity_with_ratio_square_root_of_2.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Yves Baelde • File:Academ_Study_about_a_periodic_tiling_by_regular_polygons.svg Source: https://upload.wikimedia.org/wikipedia/commons/ f/f1/Academ_Study_about_a_periodic_tiling_by_regular_polygons.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Baelde • File:Affine_space,_projective_space,_vector_space.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3a/Affine_space% 2C_projective_space%2C_vector_space.svg License: Public domain Contributors: Own work, created as per: Help:Displaying a formula: Commutative diagrams; source code below. Original artist: Nils R. Barth • File:Affine_subspace.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/8c/Affine_subspace.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Jakob.scholbach • File:Affine_transformations.ogv Source: https://upload.wikimedia.org/wikipedia/commons/3/34/Affine_transformations.ogv License: Public domain Contributors: Own work Original artist: LucasVB • File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public do- main Contributors: Own work, based off of Image:Ambox scales.svg Original artist: Dsmurat (talk · contribs) • File:An_Oasis,_or_a_Secret_Lair?.jpg Source: https://upload.wikimedia.org/wikipedia/commons/7/73/An_Oasis%2C_or_a_Secret_ Lair%3F.jpg License: CC BY 4.0 Contributors: http://www.eso.org/public/images/potw1340a/ Original artist: John Colosimo (colosi- mophotography.com)/ESO • File:Antenna-tower-collinear-et-al.jpg Source: https://upload.wikimedia.org/wikipedia/commons/a/ac/Antenna-tower-collinear-et-al. jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: Adamantios • File:Area_parallellogram_as_determinant.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/ad/Area_parallellogram_ as_determinant.svg License: Public domain Contributors: Own work, created with Inkscape Original artist: Jitse Niesen • File:AxialTiltObliquity.png Source: https://upload.wikimedia.org/wikipedia/commons/6/61/AxialTiltObliquity.png License: CC BY 3.0 Contributors: self-made by Dna-webmaster; earth-image from NASA Original artist: Dna-webmaster • File:Axonometric_projection.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/48/Axonometric_projection.svg License: Public domain Contributors: This vector image was created with Inkscape. Original artist: Yuri Raysper 304 CHAPTER 42. VECTOR SPACE

• File:Barycentric_view_of_Pluto_and_Charon_29_May-3_June_by_Ralph_in_near-true_colours.gif Source: https://upload.wikimedia. org/wikipedia/commons/2/27/Barycentric_view_of_Pluto_and_Charon_29_May-3_June_by_Ralph_in_near-true_colours.gif License: Pub- lic domain Contributors: http://www.nasa.gov/feature/pluto-and-its-moon-charon-now-in-color Original artist: NASA/JHU-APL/SWRI • File:Big-Endian.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/54/Big-Endian.svg License: Public domain Contribu- tors: Own work Original artist: R. S. Shaw • File:Boolean_functions_like_1000_nonlinearity.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/21/Boolean_functions_ like_1000_nonlinearity.svg License: Public domain Contributors: Own work Original artist: Lipedia • File:Cartesian-coordinate-system-with-circle.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/2e/Cartesian-coordinate-system-with-circle. svg License: CC-BY-SA-3.0 Contributors: ? Original artist: User 345Kai on en.wikipedia • File:Cartesian-coordinate-system.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/0e/Cartesian-coordinate-system. svg License: Public domain Contributors: Made by K. Bolino (Kbolino), based upon earlier versions. Original artist: K. Bolino • File:Cartesian_coordinate_surfaces.png Source: https://upload.wikimedia.org/wikipedia/commons/9/94/Cartesian_coordinate_surfaces. png License: CC BY 3.0 Contributors: Own work Original artist: WillowW • File:Cartesian_coordinate_system_handedness.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e2/Cartesian_coordinate_ system_handedness.svg License: CC-BY-SA-3.0 Contributors: ? Original artist: ? • File:Cartesian_coordinates_2D.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1a/Cartesian_coordinates_2D.svg Li- cense: CC-BY-SA-3.0 Contributors: ? Original artist: ? • File:Central_dilation.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a1/Central_dilation.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: KlioKlein • File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? Contributors: ? Origi- nal artist: ? • File:Compfun.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c9/Compfun.svg License: Public domain Contributors: Own work Original artist: Tlep • File:Composition_of_Inverses.png Source: https://upload.wikimedia.org/wikipedia/commons/4/4a/Composition_of_Inverses.png Li- cense: Public domain Contributors: Transferred from en.wikipedia to Commons. Original artist: Jim.belk at English Wikipedia • File:Coord_system_CA_0.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/69/Coord_system_CA_0.svg License: Pub- lic domain Contributors: Own work Original artist: Jorge Stolfi • File:Coordinate_with_Origin.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/ff/Coordinate_with_Origin.svg License: CC-BY-SA-3.0 Contributors: ? Original artist: ? • File:CubeOcathedronDualPair.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/2c/CubeOcathedronDualPair.svg Li- cense: CC BY-SA 3.0 Contributors: File:CubeOcathedronDualPair.jpg Original artist: Original rendering in JPG-Format: DanRadin

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• File:Frieze_example_p2mg.png Source: https://upload.wikimedia.org/wikipedia/commons/7/7b/Frieze_example_p2mg.png License: GFDL Contributors: File:Frieze2b.png Original artist: AndrewKepert • File:Geom_podobnost_stejnolehlest.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4f/Geom_podobnost_stejnolehlest. svg License: Public domain Contributors: Originally from cs.wikipedia; description page is/was here. 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