Linear Algebra is intended for a one-term course at the junior or senior level. It begins with an exposition of the basic theory of vector spaces and proceeds to explain the fundamental structure theorem for linear maps, including eigenvectors and eigenvalues, quadratic and hermitian forms, diagnolization of symmetric, hermitian, and unitary linear maps and matrices, triangulation, and Jordan canonical form. The book also includes a useful chapter on convex sets and the finite-dimensional Krein-Milman theorem. The presentation is aimed at the student who has already had some exposure to the elementary theory of matrices, determinants and linear maps. However the book is logically self-contained. In this new edition, many parts of the book have been rewritten and reorganized, and new exercises have been added.
Author(s): Serge Lang
Series: Undergraduate Texts in Mathematics
Edition: 3
Publisher: Springer
Year: 1987
Language: English
Pages: 308
Foreword......Page 6
Contents......Page 8
CHAPTER I Vector Spaces......Page 12
§1. Definitions......Page 13
§2. Bases......Page 21
§3. Dimension of a Vector Space......Page 26
§4. Sums and Direct Sums......Page 30
§1. The Space of Matrices......Page 34
§2. Linear Equations......Page 40
§3. Multiplication of Matrices......Page 42
§1. Mappings......Page 54
§2. Linear Mappings......Page 62
§3. The Kernel and Image of a Linear Map......Page 70
§4. Composition and Inverse of Linear Mappings......Page 77
§5. Geometric Applications......Page 83
§1. The Linear Map Associated with a Matrix......Page 92
§2. The Matrix Associated with a Linear Map......Page 93
§3. Bases, Matrices, and Linear Maps......Page 98
§1. Scalar Products......Page 106
§2. Orthogonal Bases, Positive Definite Case......Page 114
§3. Application to Linear Equations; the Rank......Page 124
§4. Bilinear Maps and Matrices......Page 129
§5. General Orthogonal Bases......Page 134
§6. The Dual Space and Scalar Products......Page 136
§7. Quadratic Forms......Page 143
§8. Sylvester's Theorem......Page 146
§1. Determinants of Order 2......Page 151
§2. Existence of Determinants......Page 154
§3. Additional Properties of Determinants......Page 161
§4. Cramer's Rule......Page 168
§5. Triangulation of a Matrix by Column Operations......Page 172
§6. Permutations......Page 174
§7. Expansion Formula and Uniqueness of Determinants......Page 179
§8. Inverse of a Matrix......Page 185
§9. The Rank of a Matrix and Subdeterminants......Page 188
§1. Symmetric Operators......Page 191
§2. Hermitian Operators......Page 195
§3. Unitary Operators......Page 199
§1. Eigenvectors and Eigenvalues......Page 205
§2. The Characteristic Polynomial......Page 211
§3. Eigenvalues and Eigenvectors of Symmetric Matrices......Page 224
§4. Diagonalization of a Symmetric Linear Map......Page 229
§5. The Hermitian Case......Page 236
§6. Unitary Operators......Page 238
§1. Polynomials......Page 242
§2. Polynomials of Matrices and Linear Maps......Page 244
§1. Existence of Triangulation......Page 248
§2. Theorem of Hamilton-Cayley......Page 251
§3. Diagonalization of Unitary Maps......Page 253
§1. The Euclidean Algorithm......Page 256
§2. Greatest Common Divisor......Page 259
§3. Unique Factorization......Page 262
§4. Application to the Decomposition of a Vector Space......Page 266
§5. Schur's Lemma......Page 271
§6. The Jordan Normal Form......Page 273
§1. Definitions......Page 279
§2. Separating Hyperplanes......Page 281
§3. Extreme Points and Supporting Hyperplanes......Page 283
§4. The Krein-Milman Theorem......Page 285
APPENDIX I Complex Numbers......Page 288
APPENDIX II Iwasawa Decomposition and Others......Page 294
Index......Page 304
