## Linear Algebra – Voyevodin

In this post, we will see the book Linear Algebra by V. V. Voyevodin. The associated problem book by by H. D. Ikramov can be seen here. This textbook is a comprehensive united course in linear algebra and analytic geometry based on lectures read by the author for many years at various institutes to future specialists in computational mathematics.
It is intended mainly for those in whose education computational mathematics is to occupy a substantial place. Much of the instruction in this speciality is connected with the traditional mathematical courses. Nevertheless the interests of computational mathematics make it necessary to introduce large enough changes in both the methods of presentation of these courses and their content.

The book was translated from the Russian by Vladimir Shokurov and was first published by Mir Publishers in 1983.

PDF | OCR | Bookmarked | 393 p.

All credits to the original uploader.

Contents

Front Cover 1
Title Page 4
Contents 5
Preface 9

PART I Vector Spaces 11 396

CHAPTER 1 Sets, Elements, Operations 11 396

1. Sets and elements 12 107
2. Algebraic operation 14 104
3. lnverse operation 18 588
4. Equivalence relation 20 95
5. Directed line segments 22 41
6. Addition of directed line segments 25 452
7. Groups 28 540
8. Rings and fields 31 85
9. Multiplication of directed line segments by a number 35 531
10. Vector spaces 38 517
11. Finite sums and products 41 21
12. Approximate calculations 44 311

CHAPTER 2 The Structure of a Vector Space 45 329

13. Linear combinations and spans 46 225
14. Linear dependence 48 79
15. Equivalent systems of vectors 51 465
16. The basis 54 146
17. Simple examples of vector spaces 56 143
18. Vector spaces of directed line segments 57 299
19. The sum and intersection of subspaces 61 113
20. The direct sum of subspaces 64 403
21. Isomorphism of vector spaces 66 90
22. Linear dependence and systems of linear equations 70 324

CHAPTER 3 Measurements in Vector Space 75 452

23. Affine coordinate systems 75 452
24. Other coordinate systems 80 429
25. Some problems 82 317
26. Scalar product 89 310
27. Euclidean space 92 67
28. Orthogonality 95 128
29. Lengths, angles, distances 99 109
30. Inclined line, perpendicular, projection 102 577

CHAPTER 4 The Volume of a System of Vectors in Vector Space 110 44

31. Euclidean isomorphism 106 533
32. Unitary spaces 107 223
33. Linear dependence and orthonormal systems 108 213
34. Vector and triple scalar products 110 171
35. Volume and oriented volume of a system of vectors 115 9
36. Geometrical and algebraic properties of a volume 118 299
37. Algebraic properties of an oriented volume 122 0
38. Permutations 125 595
39. The existence of an oriented volume 126 95
40. Determinants 128 333
41. Linear dependence and determinants 133 27
42. Calculation of determinants 136 137

CHAPTER 5 The Straight Line and the Plane in Vector Space 137 104

43. The equations of a straight line and of a plane 137 104
44. Relative positions 142 220
45. The plane in vector space 146 164
46. The straight line and the hyperplane 149 229
47. The half space 154 230

CHAPTER 6 The Limit in Vector Space 161 146

49. Metric spaces 161 146
50. Complete spaces 163 359
51. Auxiliary inequalities 166 542
52. Normed spaces 168 53
53. Convergence in the norm and coordinate convergence 170 116
54. Completeness of normed spaces 173 76
55. The limit and computational processes 175 35

PART II Linear Operators 177 197

CHAPTER 7 Matrices and Linear Operators 177 400

56. Operators 177 400
57. The vector space of operators 181 115
58. The ring of operators 183 232
59. The group of nonsingular operators 185 248
60. The matrix of an operator 188 21
61. Operations on matriees 192 139
62. Matrices and determinants 196 449
63. Change of basis 199 72
64. Equivalent and similar matrices 202 345

CHAPTER 8 The Characteristic Polynomial 205 424

65. Eigenvalues and eigenvectors 205 424
66. The characteristic polynomial 208 271
67. The polynomial ring 210 238
68. The fundamental theorem of algebra 214 318
69. Consequences of the fundamental theorem 218 5

CHAPTER 9 The Structureof a Linear Operator 223 530

70. Invariant subspaees 223 530
71. The operator polynomial 225 384
72. The triangular form 228 211
73. A direct sum of operators 229 277
74. The Jordan canonical form 232 291
75. The adjoint operator 236 326
76. The normal operator 240 329
77. Unitary and Hermitian operators 243 546
78. Operators A*A and AA* 246 282
79. Decomposition of an arbitrary operator 249 169
80. Operators in the real space 251 304
81. Matrices of a special form 254 39

CHAPTER 10 Metric Properties of an Operator 257 454

82. The continuity and boundedness of an operator 257 454
83. The norm of an operator 259 23
84. Matrix norms of an operator 263 283
85. Operator equations 266 243
86. Pseudosolutions andthe pseudoinverse operator 268 412
87. Perturbation and nonsingularity of an operator 271 16
88. Stable solution of equations 275 276
89. Perturbation and eigenvalues 280 104

PART III Bilinear Forms 283 296

CHAPTER 11 Bilinear and Quadratic Forms 284 107

90. General properties of bilinearand quadratic forms 284 107
91. The matrices of bilinear and quadratic forms 290 359
92. Reduction to canonical form 296 331
93. Congruence and matrix decompositions 304 368
94. Symmetric bilinear forms 309 240
95. Second degree hypersurfaces 316 273
96. Second degree curves 321 109
97. Second degree surfaces 328 331

CHAPTER 12 Bilinear Metric Spaces 333 227

98. The Gram matrix and determinant 334 4
99. Nonsingular subspaces 340 248
100. Orthogonality in bases 344 509
101. Operators and bilinear forms 350 352
102. Bilinear metric Isomorphism 355 480

CHAPTER 13 Bilinear Forms in Computational Processes 358 213

103. Orthogonalization processes 358 213
104. Orthogonalizatio of a power sequence 363 322
105. Methods of conjugate directions 368 289
106. Main variants 374 71
107. Operator equations and pseudoduality 377 313
108. Bilinear forms in spectral problems 382 248

Conclusion 388 498

INDEX 390 336 1. https://ybaljawordpress.com says: