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# Finite-Dimensional Vector Spaces by Paul R. Halmos (2nd Edition)

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Originally published in 1958, Finite-Dimensional Vector Spaces by Prof. Paul R. Halmos (2nd Edition) is a classic reference book on Linear Algebra, especially if you are looking for an abstract approach to linear algebra.

The purpose of this book is to treat linear transformations on finite-dimensional vector spaces by the methods of more general theories. The idea is to emphasize the simple geometric notions common to many parts of mathematics and its applications, and to do so in a language that gives away the trade secrets and tells the student what is in the back of the minds of people proving theorems about integral equations and Hilbert spaces.

Except for an occasional reference to undergraduate mathematics the book is self-contained and may be read by anyone who is trying to get a feeling for the linear problems usually discussed in courses on matrix theory or "higher" algebra. The algebraic, coordinate-free methods do not lose power and elegance by specialization to a finite number of dimensions, and they are as elementary as the classical coordinatized treatment.

About the author -- Excerpt from the Wikipedia:

Paul Richard Halmos (1916 - 2006) was a Hungarian-Jewish-born American mathematician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as a great mathematical expositor.

Contents Covered:

• Preface
• Spaces
• 1. Fields
• 2. Vector spaces
• 3. Examples
• 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
• Transformations
• 32. Linear transformations
• 33. Transformations as vectors
• 34. Products
• 35. Polynomials
• 36. Inverses
• 37. Matrices
• 38. Matrices of transformations
• 39. Invariance
• 40. Reducibility
• 41. Projections
• 42. Combinations of projections
• 43. Projections and invariance
• 46. Change of basis
• 47. Similarity
• 48. Quotient transformations
• 49. Range and null-space
• 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
• Orthogonality
• 59. Inner products
• 60. Complex 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
• 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
• 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
• Index of Terms
• Index of Symbols
 Format: PDF Digital Reprint, e-Facsimile No. of Pages: 207 Page Size: A4 (210mm × 297mm) Download Size: 25.5 MB

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