دانلود کتاب Linear Algebra

فهرست مطالب :

Chapter 1

DETERMINANTS 1
1.1. Number Fields 1
1.2. Problems of the Theory of Systems of Linear Equations 3
1.3. Determinants of Order a 5
1.4. Properties of Determinants 8
1.5. Co-factors and Minors RB
1.6. Practical Evaluation of Determinants t6
1.7. Cramer\'s Rule (8
1.8. Minors of Arbitrary Order. Laplace\'s Theorem 20
1.9. Linear Dependence between Columns 23
Problems 28


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Chapter 2

LINEAR SPACES 31
2.1. Definitions 31
2.2. Linear Dependence 36
2.3. Bases, Components, Dimension 38
2.4. Subspaces 42
2.5. Linear Manifolds 49
2.6. Hyperplane 51
2.7. Morphisms of Linear Spaces 33
Problems 56

Chapter 3

SYSTEMS OF LINEAR EQUATIONS 58
3.1. More on the Rank of a Matrix 8
3.2. Nontrivial Compatibility of 0 Homogeneous Linear System 60.
3.3. The Compatibility Condition for a General Linear System 61
3.4. The General Solution of a Linear System 63
3.5. Geometric Properties of the Solution Space 65
3.6. Methods for Calculating the Rank of a Matrix 67
Problems 71

Chapter 4

LINEAR FUNCTIONS OF A VECTOR ARGUMENT 75
4.1. Linear Forms 75
4.2. Linear Operators 77
4.3. Sums and Products of Linear Operators 82
4.4. Corresponding Operations on Matrices 84
4.5. Further Properties of Matrix Multiplication 88
4.6. The Range and Null Space of a Linear Operator 93
4.7. Linear Operators Mapping a Space K_n into Itself 98
4.2. Invariant Subspaces 106
4.9. Eigenvectors and Eigenvalues 108
Problems 113

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Chapter 5

COORDINATE TRANSFORMATIONS 18
5.1. Transformation to a New Basis 118
5.2. Consecutive Transformations 120
5.3. Transformation of the Components of a Vector 121
5.4. Transformation of the Coefficients of a Linear Form 123
5.5. Transformation of the Matrix of a Linear Operator 124
5.6. Tensors 126
Problems 131

Chapter 6

THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR 133
6.1. Canonical Form of the Matrix of a Nilpotent Operator 133
6.2. Algebras. The Algebra of Polynomials 136
6.3. Canonical Form of the Matrix of an Arbitrary Operator 142
6.4. Elementary Divisors 147
6.5. Further Implications 183
6.6. The Real Jordan Canonical Form 185
6.7. Spectra, Jets and Polynomials 1460
6.8. Operator Functions and Their Matrices 169
Problems 176

Chapter 7

BILINEAR AND QUADRATIC FORMS 179
7.1. Bilinear Forms 179
7.2. Quadratic Forms 183
7.3. Reduction of a Quadratic Form to Canonical Form 185
7.4. The Canonical Basis of a Bilinear Form 190
7.8. Construction of a Canonical Basis by Jacobi’s Method 192
7.6. Adjoint Linear Operators 196
7.7. Isomorphism of Spaces Equipped with a Bilinear Form 199
7.8. Multilinear Forms 202
7.9. Bilinear and Quadratic Forms in a Real Space 204
Problems 210

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Chapter 8

EUCLIDEAN SPACES 24
8.1. Introduction 24
8.2. Definition of a Euclidean Space 2ts
8.3. Basic Metric Concepts 216
8.4. Orthogonal Bases 222
8.5. Perpendiculars 223
8.6. The Orthogonalization Theorem 226
8.7. The Gram Determinant 230
8.8. Incompatible Systems and the Method of Least Squares 234
8.9. Adjoint Operators and Isometry 237
Problems 241


Chapter 9

UNITARY SPACES 247
9.1. Hermitian Forms 247
9.2. The Scalar Product in a Complex Space 254
9.3. Normal Operators 259
9.4. Applications to Operator Theory in Euclidean Space 263
Problems 271

Chapter 10

QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES 273
10.1. Basic Theorem on Quadratic Forms in a Euclidean Space 273
10.2. Extremal Properties of a Quadratic Form 276
10.3. Simultaneous Reduction of Two Quadratic Forms 283
10.4. Reduction of the General Equation of a Quadric Surface 287
10.5. Geometric Properties of a Quadric Surface 289
10.6. Analysis of a Quadric Surface from Its General Equation 300
10.7. Hermitian Quadratic Forms 308
Problems 310