线性代数习题集(英文版)(精)/美国数学会经典影印系列

本书是著名数学家Paul R. Halmos精心撰写的线 性代数学习辅导书。对于 每一位需要学习和使用线 性代数的人来说，本书既 可以作为“主菜”，也可以 作为“甜点”。本书可以作 为官方课程或个人学习计 划的基础学习资料。它可 以作为课程教材独立使用 ，或者如果需要，它也可 以与标准线性代数教材一 起使用，为初学者甚至是 经验丰富的学者提供富有 趣味和挑战性的材料。最 好的学习方法是做题，而 本书的目的就是让读者做 各式各样的线性代数习题 。方法是苏格拉底式的： 首先提出一个问题，然后 给出提示（如有必要）， 最后为了保险和完整起见 ，提供详细的答案。

Preface Chapter 1. Scalars 1. Double addition 2. Half double addition 3. Exponentiation 4. Complex numbers 5. Affine transformations 6. Matrix multiplication 7. Modular multiplication 8. Small operations 9. Identity elements 10. Complex inverses 11. Affine inverses 12. Matrix inverses 13. Abelian groups 14. Groups 15. Independent group axioms 16. Fields 17. Addition and multiplication in fields 18. Distributive failure 19. Finite fields Chapter 2. Vectors 20. Vector spaces 21. Examples 22. Linear combinations 23. Subspaces 24. Unions of subspaces 25. Spans 26. Equalities of spans 27. Some special spans 28. Sums of subspaces 29. Distributive subspaces 30. Total sets 31. Dependence 32. Independence Chapter 3. Bases 33. Exchanging bases 34. Simultaneous complements 35. Examples of independence 36. Independence over R and Q 37. Independence in C2 38. Vectors common to different bases 39. Bases in C3 40. Maximal independent sets 41. Complex as real 42. Subspaces of full dimension 43. Extended bases 44. Finite-dimensional subspaces 45. Minimal total sets 46. Existence of minimal total sets 47. Infinitely total sets 48. Relatively independent sets 49. Number of bases in a finite vector space 50. Direct sums 51. Quotient spaces 52. Dimension of a quotient space 53. Additivity of dimension Chapter 4. Transformations 54. Linear transformations 55. Domain and range 56. Kernel 57. Composition 58. Range inclusion and factorization 59. Transformations as vectors 60. Invertibility 61. Invertibility examples 62. Determinants: 2 X 2 63. Determinants: n X n 64. Zero-one matrices 65. Invertible matrix bases 66. Finite-dimensional invertibility 67. Matrices 68. Diagonal matrices 69. Universal commutativity 70. Invariance 71. Invariant complements 72. Projections 73. Sums of projections 74. not quite idempotence Chapter 5. Duality 75. Linear functionals 76. Dual spaces 77. Solution of equations 78. Reflexivity 79. Annihilators 80. Double annihilators 81. Adjoints 82. Adjoints of projections 83. Matrices of adjoints Chapter 6. Similarity 84. Change of basis: vectors 85. Change of basis: coordinates 86. Similarity: transformations 87. Similarity: matrices 88. Inherited similarity 89. Similarity: real and complex 90. Rank and nullity 91. Similarity and rank 92. Similarity of transposes 93. Ranks of sums 94. Ranks of products 95. Nullities of sums and products 96. Some similarities 97. Equivalence 98. Rank and equivalence Chapter 7. Canonical Forms 99. Eigenvalues 100. Sums and products of eigenvalues 101. Eigenvalues of products 102. Polynomials in eigenvalues 103. Diagonalizing permutations 104. Polynomials in eigenvalues, converse 105. Multiplicities 106. Distinct eigenvalues 107. Comparison of multiplicities 108. Triangularization 109. Complexification 110. Unipotent transformation 111. Nipotence 112. Nilpotent products 113. Nilpotent direct sums 114. Jordan form 115. Minimal polynomials 116. Non-commutative Lagrange interpolation Chapter 8. Inner Product Spaces 117. Inner products 118. Polarization 119. The Pythagorean theorem 120. The parallelogram law 121. Complete orthonormal sets 122. Schwarz inequality 123. Orthogonal complements 124. More linear functionals 125. Adjoints on inner product spaces 126. Quadratic forms 127. Vanishing quadratic forms 128. Hermitian transformations 129. Skew transformations 130. Real Hermitian forms 131. Positive transformations 132. positive inverses 133. Perpendicular projections 134. Projections on C X C 135. Projection order 136. Orthogonal projections 137. Hermitian eigenvalues 138. Distinct e