Linear Algebra

Contents ......Page 6 Preface ......Page 10 1. Sets and elements ......Page 12 2. Algebraic Operation ......Page 14 3. Inverse operation ......Page 17 4. Equivalence relation ......Page 20 5. Directed line segments ......Page 22 6. Addition of directed line segments ......Page 24 7. Groups ......Page 27 8. Rings and fields ......Page 31 9. Multiplication of directed line segments by a number ......Page 34 10. Vector spaces ......Page 37 11. Finite sums and products ......Page 41 12. Approximate calculations ......Page 44 13. Linear combinations and spans ......Page 46 14. Linear dependence ......Page 48 15. Equivalent systems of vectors ......Page 51 16. The basis ......Page 54 17. Simple examples of vector spaces ......Page 56 18. Vector spaces of directed line segments ......Page 57 19. The sum and intersection of subspaces ......Page 61 20. The direct sum of subspaces ......Page 64 21. Isomorphism of vector spaces ......Page 66 22. Linear dependence and systems of linear equations ......Page 70 23. Affine coordinate systems ......Page 75 24. Other coordinate systems ......Page 80 25. Some problems ......Page 82 26. Scalar product ......Page 89 27. Euclidean space ......Page 92 28. Orthogonality ......Page 95 29. Lengths, angles, distances ......Page 99 30. Inclined line, perpendicular, projection ......Page 102 31. Euclidean isomorphism ......Page 105 32. Unitary spaces ......Page 107 33. Linear dependence and orthonormal systems ......Page 108 34. Vector and triple scalar products ......Page 110 35. Volume and oriented volume of a system of vectors ......Page 115 36. Geometrical and algebraic properties of a volume ......Page 117 37. Algebraic properties of an oriented volume ......Page 122 38. Permutations ......Page 124 39. The existence of an oriented volume ......Page 126 40. Determinants ......Page 128 41. Linear dependence and determinants ......Page 133 42. Calculation of determinants ......Page 136 43. The equations of a straight line and of a plane ......Page 137 44. Relative positions ......Page 142 45. The plane in vector space ......Page 146 46. The straight line and the hyperplane ......Page 149 47. The half-space ......Page 154 48. Systems of linear equations ......Page 156 49. Metric spaces ......Page 161 50. Complete spaces ......Page 163 51. Auxiliary inequalities ......Page 166 52. Normed spaces ......Page 168 53. Convergence in the norm and coordinate convergence ......Page 170 54. Completeness of normed spaces ......Page 173 55. The limit and computational processes ......Page 175 56. Operators ......Page 178 57. The vector space of operators ......Page 181 58. The ring of operators ......Page 183 50. The group of nonsingular operators ......Page 185 60. The matrix of an operator ......Page 188 61. Operations on matrices ......Page 192 62. Matrices and determinants ......Page 196 63. Change of basis ......Page 199 64. Equivalent and similar matrices ......Page 202 65. Eigenvalues and eigenvectors ......Page 205 66. The characteristic polynomial ......Page 207 67. The polynomial ring ......Page 210 68. The fundamental theorem of algebra ......Page 214 69. Consequences of the fundamental theorem ......Page 218 70. Invariant subspaces ......Page 223 71. The operator polynomial ......Page 226 72. The triangular form ......Page 228 73. A direct sum of operators ......Page 229 74. The Jordan canonical form ......Page 233 75. The adjoint operator ......Page 236 76. The normal operator ......Page 241 77. Unitary and Hermitian operators ......Page 243 78. Operators A* A and A A* ......Page 247 79. Decomposition of an arbitrary operator ......Page 249 80. Operators in the real space ......Page 251 81. Matrices of a special form ......Page 254 82. The continuity and boundedness of an operator ......Page 257 83. The norm of an operator ......Page 259 84. Matrix norms of an operator ......Page 263 85. Operator equations ......Page 266 86. Pseudosolutions and the pseudoinverse operator ......Page 268 87. Perturbation and nonsingularity of an operator ......Page 271 88. Stable solution of equations ......Page 275 89. Perturbation and eigenvalues ......Page 280 90. General properties of bilinear and quadratic forms ......Page 284 91. The matrices of bilinear and quadratic forms ......Page 290 92. Reduction to canonical form ......Page 296 93. Congruence and matrix decompositions ......Page 304 94. Symmetric bilinear forms ......Page 309 95. Second-degree hypersurfaces ......Page 316 96. Second-degree curves ......Page 321 97. Second-degree surfaces ......Page 328 98. The Gram matrix and determinant ......Page 334 99. Nonsingular subspaces ......Page 340 100. Orthogonality in bases ......Page 343 101. Operators and bilinear forms ......Page 350 102. Bilinear metric isomorphism ......Page 355 103. Orthogonalization processes ......Page 358 104. Orthogonalization of a power sequence ......Page 364 105. Methods of conjugate directions ......Page 368 106. Main variants ......Page 374 107. Operator equations and pseudoduality ......Page 378 108. Bilinear forms in spectral problems ......Page 382 Conclusion ......Page 388 Index ......Page 390