| 9 | | Preface |
| 11 | Part I. | Vector Spaces |
| 11 | Chapter 1. | Sets, Elements, Operations |
| 11 | 1. | Sets and elements |
| 13 | 2. | Algebraic Operation |
| 16 | 3. | Inverse operation |
| 19 | 4. | Equivalence relation |
| 21 | 5. | Directed line segments |
| 23 | 6. | Addition of direct line segments |
| 26 | 7. | Groups |
| 30 | 8. | Rings and fields |
| 33 | 9. | Multiplication of directed line segments by a number |
| 36 | 10. | Vector spaces |
| 40 | 11. | Finite sums and products |
| 43 | 12. | Approximate calculations |
| 45 | Chapter 2. | The Structure of a Vector Space |
| 45 | 13. | Linear combinations and spans |
| 47 | 14. | Linear dependence |
| 50 | 15. | Equivalent systems of vectors |
| 53 | 16. | The basis |
| 55 | 17. | Simple examples of vector spaces |
| 56 | 18. | Vector spaces of directed line segments |
| 60 | 19. | The sum and intersection of subspaces |
| 63 | 20. | The direct sum of subspaces |
| 65 | 21. | Isomorphism of vector spaces |
| 69 | 22. | Linear dependence and systems of linear equations |
| 74 | Chapter 3. | Measurements in Vector Space |
| 74 | 23. | Affine coordinate systems |
| 79 | 24. | Other coordinate systems |
| 81 | 25. | Some problems |
| 88 | 26. | Scalar product |
| 91 | 27. | Euclidean space |
| 94 | 28. | Orthogonality |
| 98 | 29. | Lenghts, angles, distances |
| 101 | 30. | Inclined line, perpendicular, projection |
| 104 | 31. | Euclidean isomorphism |
| 106 | 32. | Unitary spaces |
| 107 | 33. | Linear dependence and orthonormal systems |
| 109 | Chapter 4. | The Volume of a System of Vectors in Vector Space |
| 109 | 34. | Vector and triple scalar products |
| 114 | 35. | Volume and oriented volume of a system of vectors |
| 116 | 36. | Geometrical and algebraic properties of a volume |
| 121 | 37. | Algebraic properties of an oeriented volume |
| 123 | 38. | Permutations |
| 125 | 39. | The existence of an oriented volume |
| 127 | 40. | Determinants |
| 132 | 41. | Linear dependence and determinants |
| 135 | 42. | Calculation of determinants |
| 136 | Chapter 5. | The Straight Line and the Plane in Vector Space |
| 136 | 43. | The equations of a straight line and of a plane |
| 141 | 44. | Relative positions |
| 145 | 45. | The plane in vector space |
| 148 | 46. | The straight line and the hyperplane |
| 153 | 47. | The half-space |
| 155 | 48. | Systems of linear equations |
| 160 | Chapter 6. | The Limit in Vector Space |
| 160 | 49. | Metric spaces |
| 162 | 50. | Complete spaces |
| 165 | 51. | Auxiliary inequalities |
| 167 | 52. | Normed spaces |
| 169 | 53. | Convergence in the norm and coordinate convergence |
| 172 | 54. | Completeness of normed spaces |
| 174 | 55. | The limit and computational processes |
| 177 | Part II. | Linear Operators |
| 177 | Chapter 7. | Matrices and Linear Operators |
| 177 | 56. | Operators |
| 180 | 57. | The vector space of operators |
| 182 | 58. | The ring of operators |
| 184 | 59. | The group of nonsingular operators |
| 187 | 60. | The matrix of an operator |
| 191 | 61. | Operations on matrices |
| 195 | 62. | Matrices and determinants |
| 198 | 63. | Change of basis |
| 201 | 64. | Equivalent and similar matrices |
| 204 | Chapter 8. | The Characteristic Polynomial |
| 204 | 65. | Eigenvalues and eigenvectors |
| 206 | 66. | he characteristic polynomial |
| 209 | 67. | The polynomial ring |
| 213 | 68. | The fundamental theorem of algebra |
| 217 | 69. | Consequences of the fundamental theorem |
| 222 | Chapter 9. | The Structure of a Linear Operator |
| 222 | 70. | Invariant subspaces |
| 225 | 71. | The operator polynomial |
| 227 | 72. | The triangular form |
| 228 | 73. | A direct sum of operators |
| 232 | 74. | The Jordan canonical form |
| 235 | 75. | he adjoint operator |
| 240 | 76. | The normal operator |
| 242 | 77. | Unitary and Hermitian operators |
| 246 | 78. | Operators A*A and AA* |
| 248 | 79. | Decomposition of an arbitrary operator |
| 250 | 80. | Operators in the real space |
| 253 | 81. | Matrices of a special form |
| 256 | Chapter 10. | Metric Properties of an Operator |
| 256 | 82. | The continuity and boundedness of an operator |
| 258 | 83. | The norm of an operator |
| 262 | 84. | Matrix norms of an operator |
| 265 | 85. | Operator equations |
| 267 | 86. | Pseudosolutions and the pseudoinverse operator |
| 270 | 87. | Perturbation and nonsingularity of an operator |
| 274 | 88. | Stable solution of equations |
| 279 | 89. | Perturbation and eigenvalues |
| 283 | Part III. | Bilinear Forms |
| 283 | Chapter 11. | Bilinear and Quadratic Forms |
| 283 | 90. | General properties of bilinear and quadric forms |
| 289 | 91. | The matrices of bilinear and quadratic forms |
| 295 | 92. | Reduction to canonical form |
| 303 | 93. | Congruence and matrix decompositions |
| 308 | 94. | Symmetric bilinear forms |
| 315 | 95. | Second-degree hypersurfaces |
| 320 | 96. | Second-degree curves |
| 327 | 97. | Second-degree surfaces |
| 333 | Chapter 12. | Bilinear Metric Spaces |
| 333 | 98. | The Gram matrix and determinant |
| 339 | 99. | Nonsingualr subspaces |
| 342 | 100. | Orthogonality in bases |
| 349 | 101. | Operators and bilinear forms |
| 354 | 102. | Bilinear metric isomorphism |
| 357 | Chapter 13. | Bilinear Forms in Computational Processes |
| 357 | 103. | Orthogonalization processes |
| 363 | 104. | Orthogonalization of a power sequence |
| 367 | 105. | Methods of conjugate directions |
| 373 | 106. | Main variants |
| 377 | 107. | Operator equations and pseudoduality |
| 381 | 108. | Bilinear forms in spectral problems |
| 387 | | Conclusion |
| 389 | | Index |
