Linear Algebra: Problems Book

Contents ......Page 6
Preface ......Page 8
1.0. Terminology and General Notes ......Page 12
1.1. Definition of Linear Space ......Page 18
1.2. Linear Dependence ......Page 20
1.3. Spans. Rank of Vector Sets ......Page 23
1.4. Basis and Dimension of Space ......Page 27
1.5. Sum and Intersection of Subspaces ......Page 30
2.0. Terminology and General Notes ......Page 34
2.1. Definition of Euclidean Space ......Page 36
2.2. Orthogonality, Orthonormal Basis, Orthogonalization Procedure ......Page 39
2.3. Orthogonal Complement, Orthogonal Sums of Subspaces ......Page 42
2.4. Lengths, Angles, Distances ......Page 46
2.5. Unitary Spaces ......Page 49
3.0. Terminology and General Notes ......Page 53
3.1. Evaluation and the Simplest Properties of Determinants ......Page 57
3.2. Minors, Cofactors and the Laplace Theorem ......Page 64
3.3. Determinants and the Volume of a Parallelepiped in a Euclidean Space ......Page 70
3.4. Computing the Determinants by the Elimination Method ......Page 75
4.0. Terminology and General Notes ......Page 82
4.1. The Rank of a Matrix ......Page 83
4.2. Planes in a Linear Space ......Page 87
4.3. Planes in a Euclidean Space ......Page 90
4.4. Homogeneous Systems of Linear Equations ......Page 93
4.5. Nonhomogeneous Systems of Linear Equations ......Page 100
5.0. Terminology and General Notes ......Page 109
5.1. The Definition of a Linear Operator, the Image and Kernel of an Operator ......Page 113
5.2. Linear Operations over Operators ......Page 118
5.3. Multiplication of Operators ......Page 120
5.4. Operations over Matrices ......Page 125
5.5. The Inverse of a Matrix ......Page 138
5.6. The Matrix of a Linear Operator, Transfer to Another Basis, Equivalent and Similar Matrices ......Page 147
6.0. Terminology and General Notes ......Page 153
6.1. Eigenvalues and Eigenvectors ......Page 154
6.2. .The Characteristic Polynomial ......Page 157
6.3. Invariant Subspaces ......Page 162
6.4. Root Subspaces and the Jordan Form ......Page 167
7.0. Terminology and General Notes ......Page 179
7.1. Conjugate Operator. Conjugate Matrix ......Page 183
7.2. Normal Operators and Matrices ......Page 188
7.3. Unitary Operators and Matrices ......Page 192
7.4. Hermitian Operators and Matrices ......Page 197
7.5. Positive-Semidefinite and Positive-Definite Operators and Matrices ......Page 202
7.6. Singular Values and the Polar Representation ......Page 209
7.7. Hermitian Decomposition ......Page 214
7.8. Pseudosolutions and Pseudoinverse Operators ......Page 217
7.9. Quadratic Forms ......Page 222
8.0. Terminology and General Notes ......Page 228
8.1. Normed Linear Space ......Page 231
8.2. Norms of Operators and Matrices ......Page 236
8.3. Matrix Norms and Systems of Linear Equations ......Page 240
8.4. Matrix Norms and Eigenvalues ......Page 245
Hints ......Page 254
Answers and Solutions ......Page 267
Index ......Page 325