Linear Algebra Problem Book -- copyright page -- Preface -- Contents -- 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 -- 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 -- 3 BASES -- 33. Exchanging bases -- 34. Simultaneous complements -- 35. Examples of independence -- 36. Independence over R and Q -- 37. Independence in C^2 -- 38. Vectors common to different bases -- 39. Bases in C^3 -- 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 -- 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 -- 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 -- 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 -- 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 transformations -- 111. Nilpotence -- 112. Nilpotent products -- 113. Nilpotent direct sums -- 114. Jordan form -- 115. Minimal polynomials -- 116. Non-commutative Lagrange interpolation -- 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 -- 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 -- Chapter 1. Scalars -- Chapter 2. Vectors -- Chapter 3. Bases -- Chapter 4. Transformations -- Chapter 5. Duality -- Chapter 6. Similarity -- Chapter 7. Canonical Forms -- Chapter 8. Inner Product Spaces -- Chapter 9. Normality -- SOLUTIONS -- Chapter 1. Scalars -- Solution 1. -- Solution 2. -- Solution 3. -- Solution 4. -- Solution 5. -- Solution 6. -- Solution 7. -- Solution 8. -- Solution 9. -- Solution 10. -- Solution 11. -- Solution 12. -- Solution 13. -- Solution 14. -- Solution 15. -- Solution 16. -- Solution 17. -- Solution 18. -- Solution 19. -- Chapter 2. Vectors -- Solution 20. -- Solution 21. -- Solution 22. -- Solution 23. -- Solution 24. -- Solution 25. -- Solution 26. -- Solution 27. -- Solution 28. -- Solution 29. -- Solution 30. -- Solution 31.

32 Solution 32. -- Chapter 3. Bases -- Solution 33. -- Solution 34. -- Solution 35. -- Solution 36. -- Solution 37. -- Solution 38. -- Solution 39. -- Solution 40. -- Solution 41. -- Solution 42. -- Solution 43. -- Solution 44. -- Solution 45. -- Solution 46. -- Solution 47. -- Solution 48. -- Solution 49. -- Solution 50. -- Solution 51. -- Solution 52. -- Solution 53. -- Chapter 4. Transformations -- Solution 54. -- Solution 55. -- Solution 56. -- Solution 57. -- Solution 58. -- Solution 59. -- Solution 60. -- Solution 61. -- Solution 62. -- Solution 63. -- Solution 64. -- Solution 65. -- Solution 67. -- Solution 68. -- Solution 69. -- Solution 70. -- Solution 71. -- Solution 72. -- Solution 73. -- Solution 74. -- Chapter 5. Duality -- Solution 75. -- Solution 76. -- Solution 77. -- Solution 78. -- Solution 79. -- Solution 80. -- Solution 81. -- Solution 82. -- Solution 83. -- Chapter 6. Similarity -- Solution 84. -- Solution 85. -- Solution 86. -- Solution 87. -- Solution 88. -- Solution 89. -- Solution 90. -- Solution 91. -- Solution 92. -- Solution 93. -- Solution 94. -- Solution 95. -- Solution 96. -- Solution 97. -- Solution 98. -- Chapter 7. Canonical Forms -- Solution 99. -- Solution 100. -- Solution 101. -- Solution 102. -- Solution 103. -- Solution 104. -- Solution 105. -- Solution 106. -- Solution 107. -- Solution 108. -- Solution 109. -- Solution 110. -- Solution 111. -- Solution 112. -- Solution 113. -- Solution 114. -- Solution 115. -- Solution 116. -- Chapter 8. Inner Product Spaces -- Solution 117. -- Solution 118. -- Solution 119. -- Solution 120. -- Solution 121. -- Solution 122. -- Solution 123. -- Solution 124. -- Solution 125. -- Solution 126. -- Solution 127. -- Solution 128. -- Solution 129. -- Solution 130. -- Solution 131. -- Solution 132. -- Solution 133. -- Solution 134. -- Solution 135. -- Solution 136. -- Solution 137.

Solution 138. -- Chapter 9. Normality -- Solution 139. -- Solution 140. -- Solution 141. -- Solution 142. -- Solution 143. -- Solution 144. -- Solution 145. -- Solution 146. -- Solution 147. -- Solution 148. -- Solution 149. -- Solution 150. -- Solution 151. -- Solution 152. -- Solution 153. -- Solution 154. -- Solution 155. -- Solution 156. -- Solution 157. -- Solution 158. -- Solution 159. -- Solution 160. -- Solution 161. -- Solution 162. -- Solution 163. -- Solution 164. -- Blank Page.