Voyevodin, V. V. & Shokurov, Vladimir (tr.)

Linear Algebra

Mir Publishers

[Matematica]

Mosca 1983

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[i][c] INDICE:

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 |

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