Exercises and Problems in Linear Algebra (Perfect Revision Tool)

Contents PREFACE vii Part 1. MATRICES AND LINEAR EQUATIONS 1 Chapter 1. SYSTEMS OF LINEAR EQUATIONS 3 1.1. Background 3 1.2. Exercises 4 1.3. Problems 7 1.4. Answers to Odd-Numbered Exercises 8 Chapter 2. ARITHME ...

Contents PREFACE vii Part 1. MATRICES AND LINEAR EQUATIONS 1 Chapter 1. SYSTEMS OF LINEAR EQUATIONS 3 1.1. Background 3 1.2. Exercises 4 1.3. Problems 7 1.4. Answers to Odd-Numbered Exercises 8 Chapter 2. ARITHMETIC OF MATRICES 9 2.1. Background 9 2.2. Exercises 10 2.3. Problems 12 2.4. Answers to Odd-Numbered Exercises 14 Chapter 3. ELEMENTARY MATRICES; DETERMINANTS 15 3.1. Background 15 3.2. Exercises 17 3.3. Problems 22 3.4. Answers to Odd-Numbered Exercises 23 Chapter 4. VECTOR GEOMETRY IN R n 25 4.1. Background 25 4.2. Exercises 26 4.3. Problems 28 4.4. Answers to Odd-Numbered Exercises 29 Part 2. VECTOR SPACES 31 Chapter 5. VECTOR SPACES 33 5.1. Background 33 5.2. Exercises 34 5.3. Problems 37 5.4. Answers to Odd-Numbered Exercises 38 Chapter 6. SUBSPACES 39 6.1. Background 39 6.2. Exercises 40 6.3. Problems 44 6.4. Answers to Odd-Numbered Exercises 45 Chapter 7. LINEAR INDEPENDENCE 47 7.1. Background 47 7.2. Exercises 49 iii iv CONTENTS 7.3. Problems 51 7.4. Answers to Odd-Numbered Exercises 53 Chapter 8. BASIS FOR A VECTOR SPACE 55 8.1. Background 55 8.2. Exercises 56 8.3. Problems 57 8.4. Answers to Odd-Numbered Exercises 58 Part 3. LINEAR MAPS BETWEEN VECTOR SPACES 59 Chapter 9. LINEARITY 61 9.1. Background 61 9.2. Exercises 63 9.3. Problems 67 9.4. Answers to Odd-Numbered Exercises 70 Chapter 10. LINEAR MAPS BETWEEN EUCLIDEAN SPACES 71 10.1. Background 71 10.2. Exercises 72 10.3. Problems 74 10.4. Answers to Odd-Numbered Exercises 75 Chapter 11. PROJECTION OPERATORS 77 11.1. Background 77 11.2. Exercises 78 11.3. Problems 79 11.4. Answers to Odd-Numbered Exercises 80 Part 4. SPECTRAL THEORY OF VECTOR SPACES 81 Chapter 12. EIGENVALUES AND EIGENVECTORS 83 12.1. Background 83 12.2. Exercises 84 12.3. Problems 85 12.4. Answers to Odd-Numbered Exercises 86 Chapter 13. DIAGONALIZATION OF MATRICES 87 13.1. Background 87 13.2. Exercises 89 13.3. Problems 91 13.4. Answers to Odd-Numbered Exercises 92 Chapter 14. SPECTRAL THEOREM FOR VECTOR SPACES 93 14.1. Background 93 14.2. Exercises 94 14.3. Answers to Odd-Numbered Exercises 96 Chapter 15. SOME APPLICATIONS OF THE SPECTRAL THEOREM 97 15.1. Background 97 15.2. Exercises 98 15.3. Problems 102 15.4. Answers to Odd-Numbered Exercises 103 Chapter 16. EVERY OPERATOR IS DIAGONALIZABLE PLUS NILPOTENT 105 CONTENTS v 16.1. Background 105 16.2. Exercises 106 16.3. Problems 110 16.4. Answers to Odd-Numbered Exercises 111 Part 5. THE GEOMETRY OF INNER PRODUCT SPACES 113 Chapter 17. COMPLEX ARITHMETIC 115 17.1. Background 115 17.2. Exercises 116 17.3. Problems 118 17.4. Answers to Odd-Numbered Exercises 119 Chapter 18. REAL AND COMPLEX INNER PRODUCT SPACES 121 18.1. Background 121 18.2. Exercises 123 18.3. Problems 125 18.4. Answers to Odd-Numbered Exercises 126 Chapter 19. ORTHONORMAL SETS OF VECTORS 127 19.1. Background 127 19.2. Exercises 128 19.3. Problems 129 19.4. Answers to Odd-Numbered Exercises 131 Chapter 20. QUADRATIC FORMS 133 20.1. Background 133 20.2. Exercises 134 20.3. Problems 136 20.4. Answers to Odd-Numbered Exercises 137 Chapter 21. OPTIMIZATION 139 21.1. Background 139 21.2. Exercises 140 21.3. Problems 141 21.4. Answers to Odd-Numbered Exercises 142 Part 6. ADJOINT OPERATORS 143 Chapter 22. ADJOINTS AND TRANSPOSES 145 22.1. Background 145 22.2. Exercises 146 22.3. Problems 147 22.4. Answers to Odd-Numbered Exercises 148 Chapter 23. THE FOUR FUNDAMENTAL SUBSPACES 149 23.1. Background 149 23.2. Exercises 151 23.3. Problems 155 23.4. Answers to Odd-Numbered Exercises 157 Chapter 24. ORTHOGONAL PROJECTIONS 159 24.1. Background 159 24.2. Exercises 160 vi CONTENTS 24.3. Problems 163 24.4. Answers to Odd-Numbered Exercises 164 Chapter 25. LEAST SQUARES APPROXIMATION 165 25.1. Background 165 25.2. Exercises 166 25.3. Problems 167 25.4. Answers to Odd-Numbered Exercises 168 Part 7. SPECTRAL THEORY OF INNER PRODUCT SPACES 169 Chapter 26. SPECTRAL THEOREM FOR REAL INNER PRODUCT SPACES 171 26.1. Background 171 26.2. Exercises 172 26.3. Problem 174 26.4. Answers to the Odd-Numbered Exercise 175 Chapter 27. SPECTRAL THEOREM FOR COMPLEX INNER PRODUCT SPACES 177 27.1. Background 177 27.2. Exercises 178 27.3. Problems 181 27.4. Answers to Odd-Numbered Exercises 182 Bibliography 183 Index 185