.

0 (hodnocen0 x )

BK

[Washington] : Mathematical Association of America, c1995

xiii, 336 s. ; 24 cm

ISBN 0-88385-322-1 (brož.)

The Dolciani mathematical expositions ; no. 16

000184898

CONTENTS // 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 // nK 1 // ele // 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 // X // LINEAR ALGEBRA PROBLEM BOOK // 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...39 // 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 ...51 // 54. Linear transformations // 55. Domain and range // 56. Kernel // 57. Composition // 58. Range inclusion and factorization // (n) // 59. Transformations as vectors // 60. Invertibility // 61. Invertibility examples // 62. Determinants: 2x2 // CONTENTS // xi // 63. Determinants: n X n // 64. Zero-one matrices // 65. Invertible matrix bases // 66. Finite-dimensional invertibility // 67. Matrices // 68. Diagonal] // (J // atrices // 69. Universal commutativity // 70. Invariance // 71. Invariant complements // 72. Projections // 73. // Sums of projections // 74. not quite idempotence // Chapter 5. Duality...85 // 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...97 // 84. Change of basis: vectors // 85. Change of basis: coordinates // • • • • // QC-OQC) // 00 00 00 00 // Similarity: transformations // Similarity: matrices // Inherited similarity // Similarity: real and complex // 90. Rank and nullity // 91. Similarity and rank // 92. Similarity of transposes // 93. Ranks of su // ii // s // 94. Ranks of products // 95. Nullities of sums and products // 96. Some similarities // 97. Equivalence // 98. Rank and equivalence // LINEAR ALGEBRA PROBLEM BOOK // g. // mmmie // Chapter 7. Canonical Forms...107 // 99. Eigenvalues

// 8 // 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 ...129 // 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 // CONTENTS // • • • // Xlll // 137. Hermitian eigenvalues // 138. Distinct eigenvalues // Chapter 9. Normality...149 // 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. // 148. // 149. // 150. // 151. // Normal diagonalizability // Normal commutativity // Adjoint commutativity // Adjoint intertwining // 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 ...169 // Solutions: // Chapter 1 ...185 // Chapter 2...204 // Chapter 3 ...216 // Chapter 4...228 // Chapter 5 ...252 // Chapter 6...259 // Chapter 7...277 // Chapter 8...296 // Chapter 9... 310