[i][c]
Voyevodin, V. V. & Shokurov, Vladimir (tr.)
Linear Algebra
Mir Publishers

Mosca 1983 #matematica
ig01#matematica
ig02#matematica

[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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