Book:Paul R. Halmos/Finite-Dimensional Vector Spaces

This book is part of Springer's Undergraduate Texts in Mathematics series.

Subject Matter

 * Linear Algebra
 * Vector Spaces
 * Linear Maps
 * Inner Products

Contents

 * Preface


 * I. SPACES
 * 1. Fields
 * 2. Vector spaces
 * 3. Examples
 * 4. Comments
 * 5. Linear dependence
 * 6. Linear combinations
 * 7. Bases
 * 8. Dimension
 * 9. Isomorphism
 * 10. Subspaces
 * 11. Calculus of subspaces
 * 12. Dimension of a subspace
 * 13. Dual spaces
 * 14. Brackets
 * 15. Dual bases
 * 16. Reflexivity
 * 17. Annihilators
 * 18. Direct sums
 * 19. Dimension of a direct sum
 * 20. Dual of a direct sum
 * 21. Quotient spaces
 * 22. Dimension of a quotient space
 * 23. Bilinear forms
 * 24. Tensor products
 * 25. Product bases
 * 26. Permutations
 * 27. Cycles
 * 28. Parity
 * 29. Multilinear forms
 * 30. Alternating forms
 * 31. Alternating forms of maximal degree


 * II. TRANSFORMATIONS
 * 32. Linear transformations
 * 33. Transformations as vectors
 * 34. Products
 * 35. Polynomials
 * 36. Inverses
 * 37. Matrices
 * 38. Matrices of transformations
 * 39. Invariance
 * 40. Reducibilty
 * 41. Projections
 * 42. Combinations of projections
 * 43. Projections and invariance
 * 44. Adjoints
 * 45. Adjoints of projections
 * 46. Change of basis
 * 47. Similarity
 * 48. Quotient transformations
 * 49. Range and nullspace
 * 50. Rank and nullity
 * 51. Transformations of rank one
 * 52. Tensor products of transformations
 * 53. Determinants
 * 54. Proper values
 * 55. Multiplicity
 * 56. Triangular form
 * 57. Nilpotence
 * 58. Jordan form


 * III. ORTHOGONALITY
 * 59. Inner products
 * 60. Comples inner products
 * 61. Inner product spaces
 * 62. Orthogonality
 * 63. Completeness
 * 64. Schwarz's inequality
 * 65. Complete orthonormal sets
 * 66. Projection theorem
 * 67. Linear functionals
 * 68. Parentheses versus brackets
 * 69. Natural isomorphisms
 * 70. Self-adjoint transformations
 * 71. Polarization
 * 72. Positive transformations
 * 73. Isometries
 * 74. Change of orthonormal basis
 * 75. Perpendicular projections
 * 76. Combinations of perpendicular projections
 * 77. Complexification
 * 78. Characterization of spectra
 * 79. Spectral theorem
 * 80. Normal transformations
 * 81. Orthogonal transformations
 * 82. Functions of transformations
 * 83. Polar decomposition
 * 84. Commutativity
 * 85. Self-adjoint transformations of rank one


 * IV. ANALYSIS
 * 86. Convergence of vectors
 * 87. Norm
 * 88. Expressions for the norm
 * 89. Bounds of a self-adjoint transformation
 * 90. Minimax principle
 * 91. Convergence of linear transformations
 * 92. Ergodic theorem
 * 93. Power series


 * Appendix. HILBERT SPACE
 * Recommended Reading
 * Index of Terms
 * Index of Symbols