Lang S. Linear algebra. - 3rd ed. - New York; London: Springer-Verlag, 1987. - 296 p.: ill. - ISBN 0-387-96412-6  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Contents

CHAPTER I Vector Spaces ......................................... 1 §1. Definitions ................................................. 2 §2. Bases ...................................................... 10 §3. Dimension of a Vector Space ................................ 15 §4. Sums and Direct Sums ....................................... 19 CHAPTER II Matrices ............................................ 23 §1. The Space of Matrices ...................................... 23 §2. Linear Equations ........................................... 29 §3. Multiplication of Matrices ................................. 31 CHAPTER III Linear Mappings .................................... 43 §1. Mappings ................................................... 43 §2. Linear Mappings ............................................ 51 §3. The Kernel and Image of a Linear Map ....................... 59 §4. Composition and Inverse of Linear Mappings ................. 66 §5. Geometric Applications ..................................... 72 CHAPTER IV Linear Maps and Matrices ............................ 81 §1. The Linear Map Associated with a Matrix .................... 81 §2. The Matrix Associated with a Linear Map .................... 82 §3. Bases, Matrices, and Linear Maps ........................... 87 CHAPTER V Scalar Products and Orthogonality .................... 95 §1. Scalar Products ............................................ 95 §2. Orthogonal Bases, Positive Definite Case .................. 103 §3. Application to Linear Equations; the Rank ................. 113 §4. Bilinear Maps and Matrices ................................ 118 §5. General Orthogonal Bases .................................. 123 §6. The Dual Space and Scalar Products ........................ 125 §7. Quadratic Forms ........................................... 132 §8. Sylvester's Theorem ....................................... 135 CHAPTER VI Determinants ....................................... 140 §1. Determinants of Order 2 ................................... 140 §2. Existence of Determinants ................................. 143 §3. Additional Properties of Determinants ..................... 150 §4. Cramer's Rule ............................................. 157 §5. Triangulation of a Matrix by Column Operations ............ 161 §6. Permutations .............................................. 163 §7. Expansion Formula and Uniqueness of Determinants .......... 168 §8. Inverse of a Matrix ....................................... 174 §9. The Rank of a Matrix and Subdeterminants .................. 177 CHAPTER VII Symmetric, Hermitian, and Unitary Operators ....... 180 §1. Symmetric Operators ....................................... 180 §2. Hermitian Operators ....................................... 184 §3. Unitary Operators ......................................... 188 CHAPTER VIII Eigenvectors and Eigenvalues ..................... 194 §1. Eigenvectors and Eigenvalues .............................. 194 §2. The Characteristic Polynomial ............................. 200 §3. Eigenvalues and Eigenvectors of Symmetric Matrices ........ 213 §4. Diagonalization of a Symmetric Linear Map ................. 218 §5. The Hermitian Case ........................................ 225 §6. Unitary Operators ......................................... 227 CHAPTER IX Polynomials and Matrices ........................... 231 §1. Polynomials ............................................... 231 §2. Polynomials of Matrices and Linear Maps ................... 233 CHAPTER X Triangulation of Matrices and Linear Maps ........... 237 §1. Existence of Triangulation ................................ 237 §2. Theorem of Hamilton-Cayley ................................ 240 §3. Diagonalization of Unitary Maps ........................... 242 CHAPTER XI Polynomials and Primary Decomposition .............. 245 §1. The Euclidean Algorithm ................................... 245 §2. Greatest Common Divisor ................................... 248 §3. Unique Factorization ...................................... 251 §4. Application to the Decomposition of a Vector Space ........ 255 §5. Schur's Lemma ............................................. 260 §6. The Jordan Normal Form .................................... 262 CHAPTER XII Convex Sets ....................................... 268 §1. Definitions ............................................... 268 §2. Separating Hyperplanes .................................... 270 §3. Extreme Points and Supporting Hyperplanes ................. 272 §4. The Krein-Milman Theorem .................................. 274 APPENDIX I Complex Numbers ................................... 277 APPENDIX II Iwasawa Decomposition and Others .................. 283 Index ......................................................... 293