FUNDAMENTALS OF LINEAR ALGEBRA James B. Carrell [email protected] (July, 2005) 2 Contents 1 Introduction 11 2 Linear Equations and Matrices 15 2.1 Linear equations: the beginning of algebra . 15 2.2 Matrices . 18 2.2.1 Matrix Addition and Vectors . 18 2.2.2 Some Examples . 19 2.2.3 Matrix Product . 21 2.3 Reduced Row Echelon Form and Row Operations . 23 2.4 Solving Linear Systems via Gaussian Reduction . 26 2.4.1 Row Operations and Equivalent Systems . 26 2.4.2 The Homogeneous Case . 27 2.4.3 The Non-homogeneous Case . 29 2.4.4 Criteria for Consistency and Uniqueness . 31 2.5 Summary . 36 3 More Matrix Theory 37 3.1 Matrix Multiplication . 37 3.1.1 The Transpose of a Matrix . 39 3.1.2 The Algebraic Laws . 40 3.2 Elementary Matrices and Row Operations . 43 3.2.1 Application to Linear Systems . 46 3.3 Matrix Inverses . 49 3.3.1 A Necessary and Sufficient Condition for Existence . 50 3.3.2 Methods for Finding Inverses . 51 3.3.3 Matrix Groups . 53 3.4 The LP DU Factorization . 59 3.4.1 The Basic Ingredients: L, P, D and U . 59 3.4.2 The Main Result . 61 3 4 3.4.3 Further Uniqueness in LP DU . 65 3.4.4 Further Uniqueness in LP DU . 66 3.4.5 The symmetric LDU decomposition . 67 3.4.6 LP DU and Reduced Row Echelon Form . 68 3.5 Summary . 73 4 Fields and Vector Spaces 75 4.1 What is a Field? . 75 4.1.1 The Definition of a Field . 75 4.1.2 Arbitrary Sums and Products . 79 4.1.3 Examples . 80 4.1.4 An Algebraic Number Field . 81 4.2 The Integers Modulo a Prime p . 84 4.2.1 A Field with Four Elements . 87 4.2.2 The Characteristic of a Field . 88 4.2.3 Connections With Number Theory . 89 4.3 The Field of Complex Numbers . 93 4.3.1 The Construction . 93 4.3.2 The Geometry of C ................... 95 4.4 Vector Spaces . 98 4.4.1 The Notion of a Vector Space . 98 4.4.2 Examples . 100 4.5 Inner Product Spaces . 105 4.5.1 The Real Case . 105 4.5.2 Orthogonality . 106 4.5.3 Hermitian Inner Products . 109 4.6 Subspaces and Spanning Sets . 112 4.6.1 The Definition of a Subspace . 112 4.7 Summary . 117 5 Finite Dimensional Vector Spaces 119 5.1 The Notion of Dimension . 119 5.1.1 Linear Independence . 120 5.1.2 The Definition of a Basis . 122 5.2 Bases and Dimension . 126 5.2.1 The Definition of Dimension . 126 5.2.2 Examples . 127 5.2.3 The Dimension Theorem . 128 5.2.4 An Application . 130 5.2.5 Examples . 130 5 5.3 Some Results on Matrices . 135 5.3.1 A Basis of the Column Space . 135 5.3.2 The Row Space of A and the Ranks of A and AT . 136 5.3.3 The Uniqueness of Row Echelon Form . 138 5.4 Intersections and Sums of Subspaces . 141 5.4.1 Intersections and Sums . 141 5.4.2 The Hausdorff Intersection Formula . 142 5.4.3 Direct Sums of Several Subspaces . 144 5.4.4 External Direct Sums . 146 5.5 Vector Space Quotients . 148 5.5.1 Equivalence Relations and Quotient Spaces . 148 5.5.2 Cosets . 149 5.6 Summary . 154 6 Linear Coding Theory 155 6.1 Linear Codes . 155 6.1.1 The Notion of a Code . 156 6.1.2 Linear Codes Defined by Generating Matrices . 157 6.1.3 The International Standard Book Number . 158 6.2 Error-Correcting Codes . 162 6.2.1 Hamming Distance . 162 6.2.2 The Key Result . 164 6.3 Codes With Large Minimal Distance . 166 6.3.1 Hadamard Codes . 166 6.3.2 A Maximization Problem . 167 6.4 Perfect Linear Codes . 170 6.4.1 A Geometric Problem . 170 6.4.2 How to Test for Perfection . 171 6.4.3 The Existence of Binary Hamming Codes . 172 6.4.4 Perfect Codes and Cosets . 173 6.4.5 The hat problem . 175 6.4.6 The Standard Decoding Table . 176 6.5 Summary . 182 7 Linear Transformations 183 7.1 Definitions and Examples . 183 7.1.1 Mappings . 183 7.1.2 The Definition of a Linear Transformation . 184 7.1.3 Some Examples . 184 7.1.4 General Properties of Linear Transformations . 186 6 7.2 Linear Transformations on Fn and Matrices . 190 7.2.1 Matrix Linear Transformations . 190 7.2.2 Composition and Multiplication . 192 2 7.2.3 An Example: Rotations of R . 193 n 7.3 Geometry of Linear Transformations on R . 196 7.3.1 Transformations of the Plane . 196 7.3.2 Orthogonal Transformations . 197 7.4 The Matrix of a Linear Transformation . 202 B 7.4.1 The Matrix MB0 (T ) . 202 7.4.2 Coordinates With Respect to a Basis . 203 7.4.3 Changing Basis for the Matrix of a Linear Transfor- mation . 208 7.5 Further Results on Linear Transformations . 214 7.5.1 The Kernel and Image of a Linear Transformation . 214 7.5.2 Vector Space Isomorphisms . 216 7.5.3 Complex Linear Transformations . 216 7.6 Summary . 224 8 An Introduction to the Theory of Determinants 227 8.1 The Definition of the Determinant . 227 8.1.1 The 1 × 1 and 2 × 2 Cases . 228 8.1.2 Some Combinatorial Preliminaries . 228 8.1.3 The Definition of the Determinant . 231 8.1.4 Permutations and Permutation Matrices . 232 8.1.5 The Determinant of a Permutation Matrix . 233 8.2 Determinants and Row Operations . 236 8.2.1 The Main Result . 237 8.2.2 Consequences . 240 8.3 Some Further Properties of the Determinant . 244 8.3.1 The Laplace Expansion . 244 8.3.2 The Case of Equal Rows . 246 8.3.3 Cramer’s Rule . 247 8.3.4 The Inverse of a Matrix Over Z . 248 8.3.5 A Characterization of the Determinant . 248 8.3.6 The determinant of a linear transformation . 249 8.4 Geometric Applications of the Determinant . 251 8.4.1 Cross and Vector Products . 251 8.4.2 Determinants and Volumes . 252 8.4.3 Lewis Carroll’s identity . 253 8.5 A Concise Summary of Determinants . 255 7 8.6 Summary . 256 9 Eigentheory 257 9.1 Dynamical Systems . 257 9.1.1 The Fibonacci Sequence . 258 9.1.2 The Eigenvalue Problem . 259 9.1.3 Fibonacci Revisted . 261 9.1.4 An Infinite Dimensional Example . 261 9.2 Eigentheory: the Basic Definitions . 263 9.2.1 Eigenpairs for Linear Transformations and Matrices . 263 9.2.2 The Characteristic Polynomial . 263 9.2.3 Further Remarks on Linear Transformations . 265 9.2.4 Formulas for the Characteristic Polynomial . 266 9.3 Eigenvectors and Diagonalizability . 274 9.3.1 Semi-simple Linear Transformations and Diagonaliz- ability . 274 9.3.2 Eigenspaces . 275 9.4 When is a Matrix Diagonalizable? . 280 9.4.1 A Characterization . 280 9.4.2 Do Non-diagonalizable Matrices Exist? . 282 9.4.3 Tridiagonalization of Complex Matrices . 283 9.4.4 The Cayley-Hamilton Theorem . 285 9.5 The Exponential of a Matrix . 291 9.5.1 Powers of Matrices . 291 9.5.2 The Exponential . 291 9.5.3 Uncoupling systems . 292 9.6 Summary . 296 n n 10 The Inner Product Spaces R and C 297 10.1 The Orthogonal Projection on a Subspace . 297 10.1.1 The Orthogonal Complement of a Subspace . 298 10.1.2 The Subspace Distance Problem . 299 10.1.3 The.