Linear Algebra: Problems Book | H. D. Ikramov

It is a basic requirement that any problem book should contain a sufficient number of useful and comprehensive problems for seminar classes, home-assignments, tests and examinations. The author hopes that this requirement has been fulfilled. Moreover, he has attempted to supply the strongest students with a material for personal study, and to lead them to problems currently faced in computational algebra.

PDF | OCR | Bookmarked | 330 p.

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Contents

Front Cover 1 ,-7
Title Page 4 ,-72
Contents 6 ,-161
Preface 8 ,-197
CHAPTER 1 Linear Spaces 12 42
1.0. Terminology and General Notes 12 ,42
1.1. Definition of Linear Space 18 ,457
1.2. Linear Dependence 20 ,372
1.3. Spans. Rank of Vector Sets 23 ,75
1.4. Basis and Dimension of Space 27 ,-144
1.5. Sum and Intersection of Subspaces 30 ,281
CHAPTER 2 Euclidean and Unitary Spaces 34 45
2.0. Terminology .and General Notes 34 ,45
2.1. Definition of Euclidean Space 36 ,405
2.2. Orthogonality, Orthonormal Basis, Orthogonalization Procedure 39 ,238
2.3. Orthogonal Complement, Orthogonal Sums of Subspaces 42 ,202
2.4. Lengths, Angles, Distances 46 ,232
2.5. Unitary Spaces 49 ,392
CHAPTER 3 Determinants 53 -105
3.0. Terminology and General Notes 53 ,-105
3.1. Evaluation and the Simplest Properties of Determinants 57 ,389
3.2. Minors, Cofactors and the Laplace Theorem 64 ,297
3.3. Determinants and the Volume of a Parallelepiped in a Euclidean Space 70 ,291
3.4. Computing the Determinants by the Elimination Method 75 ,72
CHAPTER 4 Systems of Linear Equations 82 -298
4.0. Terminology and General Notes 82 ,130
4.1. The Rank of a Matrix 83 ,526
4.2. Planes in a Linear Space 87 ,438
4.3. Planes in a Euclidean Space 90 ,170
4.4. Homogeneous Systems of Linear Equations 93 ,382
4.5. Nonhomogeneous Systems of Linear Equations 100 ,143
CHAPTER 5 Linear Operators and Matrices 108 598
5.0. Terminology and General Notes 109 ,85
5.1. The Definition of a Linear Operator, the Image and Kernel of an Operator 113 ,212
5.2. Linear Operations over Operators 118 ,291
5.3. Multiplication of Operators 120 ,490
5.4. Operations over Matrices 125 ,-43
5.5. The Inverse of a Matrix 138 ,147
5.6. The Matrix of a Linear Operator, Transfer to Another Basis, Equivalent and Similar Matrices 147 ,-43
CHAPTER 6 Linear Operator Structure 153 -213
6.0. Terminology and General Notes 153 ,-213
6.1. Eigenvalues and Eigenvectors 154 ,127
6.2. The Characteristic Polynomial 157 ,287
6.3. Invariant Subspaces 162 ,415
6.4. Root Subspaces and the Jordan Form 167 ,245
CHAPTER 7 Unitary Space Operators 179 -229
7.0. Terminology and General Notes 179 ,-148
7.1. Conjugate Operator. Conjugate Matrix 183 ,-174
7.2. Normal Operators and Matrices 188 ,238
7.3. Unitary Operators and Matrices 192 ,150
7.4. Hermitian Operators and Matrices 197 ,-331
7.5. Positive-Semi definite and Positive-Definite Operators and Matrices 202 ,470
7.6. Singular Values and the Polar Representation 209 ,232
7.7. Hermitian Decomposition 214 ,441
7.8. Pseudosolutions and Pseudoinverse Operators 217 ,143
7.9. Quadratic Forms 222 ,88
CHAPTER 8 Metric Problems in Linear Space 228 -331
8.0. Terminology and General Notes 228 ,-49
8.1. Normed Linear Space 231 ,274
8.2. Norms of Operators and Matrices 236 ,-112
8.3. Matrix Norms and Systems of Linear Equations 240 ,611
8.4. Matrix Norms and Eigenvalues 245 ,343
Hints 254 ,-229
Answers and Solutions 267 ,-59
Index 325 ,-23