In this post, we will see the book *Linear Algebra *by V. V. Voyevodin.

About the book:

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 33