线性代数习题集（影印版）

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目录
- 前辅文
- 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 × 2
  - 63. Determinants: n × 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 × C
  - 135. Projection order
  - 136. Orthogonal projections
  - 137. Hermitian eigenvalues
  - 138. Distinct eigenvalues
- Chapter 9. Normality
  - 139. Unitary transformations
  - 140. Unitary matrices
  - 141. Unitary involutions
  - 142. Unitary triangles
  - 143. Hermitian diagonalization
  - 144. Square roots
  - 145. Polar decomposition
  - 146. Normal transformations
  - 147. Normal diagonalizability
  - 148. Normal commutativity
  - 149. Adjoint commutativity
  - 150. Adjoint intertwining
  - 151. Normal products
  - 152. Functions of transformations
  - 153. Gramians
  - 154. Monotone functions
  - 155. Reducing ranges and kernels
  - 156. Truncated shifts
  - 157. Non-positive square roots
  - 158. Similar normal transformations
  - 159. Unitary equivalence of transposes
  - 160. Unitary and orthogonal equivalence
  - 161. Null convergent powers
  - 162. Power boundedness
  - 163. Reduction and index 2
  - 164. Nilpotence and reduction
- Hints
- Solutions:
  - Chapter 1
  - Chapter 2
  - Chapter 3
  - Chapter 4
  - Chapter 5
  - Chapter 6
  - Chapter 7
  - Chapter 8
  - Chapter 9

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