Linear Algebra

Front Cover
Title Page
Contents
Preface
PART I Vector Spaces
CHAPTER 1 Sets, Elements, Operations
1. Sets and elements
2. Algebraic operation
3. lnverse operation
4. Equivalence relation
5. Directed line segments
6. Addition of directed line segments
7. Groups
Exercises
8. Rings and fields
Exercises
9. Multiplication of directed line segments by a number
Exercises
10. Vector spaees
11. Finite sums and products
12. Approximate calculations
CHAPTER 2 The Structureof a Vector Space
13. Linear combinations and spans
14. Linear dependence
15. Equivalent systems of vectors
16. The basis
17. Simple examples of vector spaces
18. Vector spaces of directed line segments
19. The sum and intersection of subspaces
20. The direct sum of subspaces
21. Isomorphism of vector spaces
22. Linear dependence and systems of linear equations
CHAPTER 3 Measurements in Vector Space
23. Affine coordinate systems
24. Other coordinate systems
25. Some problems
26. Scalar product
27. Euclidean space
28. Orthogonality
29. Lengths, angles, distances
30. Inclined line, perpendicular, projection
CHAPTER 4 The Volumeof a System of Vectorsin Vector Space
31. Euclidean isomorphism
32. Unitary spaces
33. Linear dependence and orthonormal systems
34. Vector and triple scalar products
35. Volume and oriented volumeof a system of vectors
36. Geometrical and algebraic properties of a volume
37. Algebraic properties of an oriented volume
38. Permutations
39. The existenceof an oriented volume
40. Determinants
41. Linear dependence and determinants
42. Calculation of determinants
CHAPTER 5 The Straight Line and the Plane in Vector Space
43. The equations of a straight line and of a plane
44. Relative positions
45. The plane in vector space
46. The straight line and the hyperplane
47. The half-space
CHAPTER 6 The Limit in Vector Space
49. Metric spaces
50. Complete spaces
51. Auxiliary inequalities
52. Normed spaces
53. Convergence in the norm and coordinate convergence
54. Completeness of normed spaces
55. The limit and computational processes
PART II Linear Operators
CHAPTER 7 Matrices and Linear Operators
56. Operators
57. The vector space of operators
58. The ring of operators
59. The group of nonsingular operators
60. The matrix of an operator
61. Operations on matriees
62. Matrices and determinants
63. Change of basis
64. Equivalent and similar matrices
CHAPTER 8 The Characteristic Polynomial
65. Eigenvalues and eigenvectors
66. The characteristic polynomial
67. The polynomial ring
68. The fundamental theorem of algebra
69. Consequences of the fundamental theorem
CHAPTER 9 The Structureof a Linear Operator
70. Invariant subspaees
71. The operator polynomial
72. The triangular form
73. A direct sum of operators
74. The Jordan canonical form
75. The adjoint operator
76. The normal operator
77. Unitary and Hermitian operators
78. Operators A*A and AA*
79. Decomposition of an arbitrary operator
80. Operators in the real space
81. Matrices of a special form
CHAPTER 10 Metric Properties of an Operator
82. The continuity and boundedness of an operator
83. The norm of an operator
84. Matrix norms of an operator
85. Operator equations
86. Pseudosolutions andthe pseudoinverse operator
87. Perturbation and nonsingularity of an operator
88. Stable solution of equations
89. Perturbation and eigenvalues
PART III Bilinear Forms
CHAPTER 11 Bilinear and Quadratic Forms
90. General properties of bilinearand quadratic forms
91. The matrices of bilinear and quadratic forms
92. Reduction to canonical form
93. Congruence and matrix decompositions
94. Symmetric bilinear forms
95. Second-degree hypersurfaces
96. Second-degree curves
97. Second-degree surfaces
CHAPTER 12 Bilinear Metric Spaces
98. The Gram matrix and determinant
99. Nonsingular subspaces
100. Orthogonality in bases
101. Operators and bilinear forms
102. Bilinear metric Isomorphism
CHAPTER 13 Bilinear Forms in Computational Processes
103. Orthogonalization processes
104. Orthogonalizatio of a power sequence
105. Methods of conjugate directions
106. Main variants
107. Operator equations and pseudoduality
108. Bilinear forms in spectral problems
Conclusion
INDEX