1 Basic Concepts of Linear Algebra by Jim Carrell Department of Mathematics University of British Columbia 2 Chapter 1 Introductory Comments to the Student This textbook is meant to be an introduction to abstract linear algebra for first, second or third year university students who are specializing in math- ematics or a closely related discipline. We hope that parts of this text will be relevant to students of computer science and the physical sciences. While the text is written in an informal style with many elementary examples, the propositions and theorems are carefully proved, so that the student will get experince with the theorem-proof style. We have tried very hard to em- phasize the interplay between geometry and algebra, and the exercises are intended to be more challenging than routine calculations. The hope is that the student will be forced to think about the material. The text covers the geometry of Euclidean spaces, linear systems, ma- trices, fields (Q; R, C and the finite fields Fp of integers modulo a prime p), vector spaces over an arbitrary field, bases and dimension, linear transfor- mations, linear coding theory, determinants, eigen-theory, projections and pseudo-inverses, the Principal Axis Theorem for unitary matrices and ap- plications, and the diagonalization theorems for complex matrices such as the Jordan decomposition. The final chapter gives some applications of symmetric matrices positive definiteness. We also introduce the notion of a graph and study its adjacency matrix. Finally, we prove the convergence of the QR algorithm. The proof is based on the fact that the unitary group is compact. Although, most of the topics listed above are found in a standard course on linear algebra, some of the topics such as fields and linear coding theory are seldom treated in such a course. Our feeling is, however, that because 3 4 coding theory is such an important component of the gadgets we use ev- eryday, such as personal computers, CD players, modems etc., and because linear coding theory gives such a nice illustration of how the basic results of linear algebra apply, including it in a basic course is clearly appropriate. Since the vector spaces in coding theory are defined over the prime fields, the students get to see explicit situations where vector space structures which don't involve the real numbers are important. This text also improves on the standard treatment of the determinant, where either its existence in the n n case for n > 3 is simply assumed or it × is defined inductively by the Laplace expansion, an d the student is forced to believe that all Laplace expansions agree. We use the classical definition as a sum over all permutations. This allows one to give a quite easy proof of the Laplace expansion, for example. Much of this material here can be covered in a 13-15 week semester. Throughout the text, we have attempted to make the explanations clear, so that students who want to read further will be able to do so. Contents 1 Introductory Comments to the Student 3 2 Euclidean Spaces and Their Geometry 11 2.1 Rn and the Inner Product. 11 2.1.1 Vectors and n-tuples . 11 2.1.2 Coordinates . 12 2.1.3 The Vector Space Rn . 14 2.1.4 The dot product . 15 2.1.5 Orthogonality and projections . 16 2.1.6 The Cauchy-Schwartz Inequality and Cosines . 19 2.1.7 Examples . 21 2.2 Lines and planes. 24 2.2.1 Lines in Rn . 24 2.2.2 Planes in R3 . 25 2.2.3 The distance from a point to a plane . 26 2.3 The Cross Product . 31 2.3.1 The Basic Definition . 31 2.3.2 Some further properties . 32 2.3.3 Examples and applications . 33 3 Linear Equations and Matrices 37 3.1 Linear equations: the beginning of algebra . 37 3.1.1 The Coefficient Matrix . 39 3.1.2 Gaussian reduction . 40 3.1.3 Elementary row operations . 41 3.2 Solving Linear Systems . 42 3.2.1 Equivalent Systems . 42 3.2.2 The Homogeneous Case . 43 3.2.3 The Non-homogeneous Case . 45 5 6 3.2.4 Criteria for Consistency and Uniqueness . 47 3.3 Matrix Algebra . 50 3.3.1 Matrix Addition and Multiplication . 50 3.3.2 Matrices Over F2: Lorenz Codes and Scanners . 51 3.3.3 Matrix Multiplication . 53 3.3.4 The Transpose of a Matrix . 54 3.3.5 The Algebraic Laws . 55 3.4 Elementary Matrices and Row Operations . 58 3.4.1 Application to Linear Systems . 60 3.5 Matrix Inverses . 63 3.5.1 A Necessary and Sufficient for Existence . 63 3.5.2 Methods for Finding Inverses . 65 3.5.3 Matrix Groups . 67 3.6 The LP DU Decomposition . 73 3.6.1 The Basic Ingredients: L; P; D and U . 73 3.6.2 The Main Result . 75 3.6.3 The Case P = In . 77 3.6.4 The symmetric LDU decomposition . 78 3.7 Summary . 84 4 Fields and vector spaces 85 4.1 Elementary Properties of Fields . 85 4.1.1 The Definition of a Field . 85 4.1.2 Examples . 88 4.1.3 An Algebraic Number Field . 89 4.1.4 The Integers Modulo p . 90 4.1.5 The characteristic of a field . 93 4.1.6 Polynomials . 94 4.2 The Field of Complex Numbers . 97 4.2.1 The Definition . 97 4.2.2 The Geometry of C . 99 4.3 Vector spaces . 102 4.3.1 The notion of a vector space . 102 4.3.2 Inner product spaces . 105 4.3.3 Subspaces and Spanning Sets . 107 4.3.4 Linear Systems and Matrices Over an Arbitrary Field 108 4.4 Summary . 112 7 5 The Theory of Finite Dimensional Vector Spaces 113 5.1 Some Basic concepts . 113 5.1.1 The Intuitive Notion of Dimension . 113 5.1.2 Linear Independence . 114 5.1.3 The Definition of a Basis . 116 5.2 Bases and Dimension . 119 5.2.1 The Definition of Dimension . 119 5.2.2 Some Examples . 120 5.2.3 The Dimension Theorem . 121 5.2.4 Some Applications and Further Properties . 123 5.2.5 Extracting a Basis Constructively . 124 5.2.6 The Row Space of A and the Rank of AT . 125 5.3 Some General Constructions of Vector Spaces . 130 5.3.1 Intersections . 130 5.3.2 External and Internal Sums . 130 5.3.3 The Hausdorff Interesction Formula . 131 5.3.4 Internal Direct Sums . 133 5.4 Vector Space Quotients . 135 5.4.1 Equivalence Relations . 135 5.4.2 Cosets . 136 5.5 Summary . 139 6 Linear Coding Theory 141 6.1 Introduction . 141 6.2 Linear Codes . 142 6.2.1 The Notion of a Code . 142 6.2.2 The International Standard Book Number . 144 6.3 Error detecting and correcting codes . 145 6.3.1 Hamming Distance . 145 6.3.2 The Main Result . 147 6.3.3 Perfect Codes . 148 6.3.4 A Basic Problem . 149 6.3.5 Linear Codes Defined by Generating Matrices . 150 6.4 Hadamard matrices (optional) . 153 6.4.1 Hadamard Matrices . 153 6.4.2 Hadamard Codes . 154 6.5 The Standard Decoding Table, Cosets and Syndromes . 155 6.5.1 The Nearest Neighbour Decoding Scheme . 155 6.5.2 Cosets . 156 6.5.3 Syndromes . 157 8 6.6 Perfect linear codes . 161 6.6.1 Testing for perfection . 162 6.6.2 The hat problem . 163 7 Linear Transformations 167 7.1 Definitions and examples . 167 7.1.1 The Definition of a Linear Transformation . 168 7.1.2 Some Examples . 168 7.1.3 The Algebra of Linear Transformations . 170 7.2 Matrix Transformations and Multiplication . 172 7.2.1 Matrix Linear Transformations . 172 7.2.2 Composition and Multiplication . 173 7.2.3 An Example: Rotations of R2 . 174 7.3 Some Geometry of Linear Transformations on Rn . 177 7.3.1 Transformations on the Plane . 177 7.3.2 Orthogonal Transformations . 178 7.3.3 Gradients and differentials . 181 7.4 Matrices With Respect to an Arbitrary Basis . 184 7.4.1 Coordinates With Respect to a Basis . 184 7.4.2 Change of Basis for Linear Transformations . 187 7.5 Further Results on Linear Transformations . 190 7.5.1 An Existence Theorem . 190 7.5.2 The Kernel and Image of a Linear Transformation . 191 7.5.3 Vector Space Isomorphisms . 192 7.6 Summary . 197 8 An Introduction to the Theory of Determinants 199 8.1 Introduction . 199 8.2 The Definition of the Determinant . 199 8.2.1 Some comments . 199 8.2.2 The 2 2 case . 200 × 8.2.3 Some Combinatorial Preliminaries . 200 8.2.4 Permutations and Permutation Matrices . 203 8.2.5 The General Definition of det(A) . 204 8.2.6 The Determinant of a Permutation Matrix . 205 8.3 Determinants and Row Operations . 207 8.3.1 The Main Result . 208 8.3.2 Properties and consequences . 211 8.3.3 The determinant of a linear transformation . 214 8.3.4 The Laplace Expansion . 215 9 8.3.5 Cramer's rule . 217 8.4 Geometric Applications of the Determinant . 220 8.4.1 Cross and vector products . 220 8.4.2 Determinants and volumes . 220 8.4.3 Change of variables formula . ..