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