Linear algebra : vector spaces and linear transformations

This textbook is directed towards students who are familiar with matrices and their use in solving systems of linear equations. The emphasis is on the algebra supporting the ideas that make linear algebra so important, both in theoretical and practical applications. The narrative is written to bring along students who may be new to the level of abstraction essential to a working understanding of linear algebra. The determinant is used throughout, placed in some historical perspective, and defined several different ways, including in the context of exterior algebras. The text details proof of the existence of a basis for an arbitrary vector space and addresses vector spaces over arbitrary fields. It develops LU-factorization, Jordan canonical form, and real and complex inner product spaces. It includes examples of inner product spaces of continuous complex functions on a real interval, as well as the background material that students may need in order to follow those discussions. Special classes of matrices make an entrance early in the text and subsequently appear throughout. The last chapter of the book introduces the classical groups. Cover 1 Title page 4 Contents 8 List of Figures 12 Preface 14 How To Use This Book 18 Notation and Terminology 22 To the Student 24 Introduction 26 Chapter 1. Vector Spaces 28 1.1. Fields 28 1.2. Vector Spaces 33 1.3. Spanning and Linear Independence 39 1.4. Bases 43 1.5. Polynomials 49 1.6. R and C in Linear Algebra 53 Chapter 2. Linear Transformations and Subspaces 56 2.1. Linear Transformations 57 2.2. Cosets and Quotient Spaces 60 2.3. Affine Sets and Mappings 66 2.4. Isomorphism and the Rank Theorem 69 2.5. Sums, Products, and Projections 72 Chapter 3. Matrices and Coordinates 80 3.1. Matrices 80 3.2. Coordinate Vectors 86 3.3. Change of Basis 91 3.4. Vector Spaces of Linear Transformations 94 3.5. Equivalences 98 Chapter 4. Systems of Linear Equations 102 Introduction 102 4.1. The Solution Set 103 4.2. Elementary Matrices 108 4.3. Reduced Row Echelon Form 111 4.4. Row Equivalence 114 4.5. An Early Use of the Determinant 118 4.6. LU-Factorization 123 Chapter 5. Introductions 132 5.1. Dual Spaces 132 5.2. Transposition and Duality 137 5.3. Bilinear Forms, Their Matrices, and Duality 141 5.4. Linear Operators and Direct Sums 146 5.5. Groups of Matrices 151 5.6. Self-Adjoint and Unitary Matrices 155 Chapter 6. The Determinant Is a Multilinear Mapping 158 6.1. Multilinear Mappings 158 6.2. Alternating Multilinear Mappings 161 6.3. Permutations, Part I 164 6.4. Permutations, Part II 170 6.5. The Determinant 174 6.6. Properties of the Determinant 177 Chapter 7. Inner Product Spaces 182 7.1. The Dot Product: Under the Hood 182 7.2. Inner Products 187 7.3. Length and Angle 189 7.4. Orthonormal Sets 193 7.5. Orthogonal Complements 199 7.6. Inner Product Spaces of Functions 201 7.7. Unitary Transformations 207 7.8. The Adjoint of an Operator 212 7.9. A Fundamental Theorem 216 Chapter 8. The Life of a Linear Operator 224 8.1. Factoring Polynomials 224 8.2. The Minimal Polynomial 226 8.3. Eigenvalues 231 8.4. The Characteristic Polynomial 237 8.5. Diagonalizability 241 8.6. Self-Adjoint Matrices Are Diagonalizable 246 8.7. Rotations and Translations 248 Chapter 9. Similarity 256 9.1. Triangularization 256 9.2. The Primary Decomposition 259 9.3. Nilpotent Operators, Part I 267 9.4. Nilpotent Operators, Part II 270 9.5. Jordan Canonical Form 273 Chapter 10. GL_{n}(F) and Friends 280 10.1. More about Groups 281 10.2. Homomorphisms and Normal Subgroups 284 10.3. The Quaternions 289 10.4. The Special Linear Group 295 10.5. The Projective Group 299 10.6. The Orthogonal Group 306 10.7. The Unitary Group 310 10.8. The Symplectic Group 315 Appendix A. Background Review 322 A.1. Logic and Proof 322 A.2. Sets 326 A.3. Well-Definedness 331 A.4. Counting 333 A.5. Equivalence Relations 337 A.6. Mappings 340 A.7. Binary Operations 343 Appendix B. R2 and R3 346 B.1. Vectors 346 B.2. The Real Plane 349 B.3. The Complex Numbers and R2 351 B.4. Real 3-Space 353 B.5. The Dot Product 357 B.6. The Cross-Product 359 Appendix C. More Set Theory 366 C.1. Partially Ordered Sets 366 C.2. Zorn’s Lemma 368 Appendix D. Infinite Dimension 376 Bibliography 382 Index 384 Back Cover 395 A textbook directed towards students who are familiar with matrices and their use in solving systems of linear equations. The emphasis is on the algebra supporting the ideas that make linear algebra so important, both in theoretical and practical applications.