线性代数【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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Chapter 1 Systems of Linear Equations and Elementary Operations on Matrices1

1.1 Systems of Linear Equations1

1.1.1 Definition1

1.1.2 Equivalent Systems and Gaussian Elimination Method3

Exercises 1.18

1.2 Elementary Operations on Matrices9

1.2.1 Elementary Row Operations9

1.2.2 Row Echelon Forms11

1.2.3 Standard Form of a Matrix15

Exercises 1.215

Chapter 2 Determinants18

2.1 The Determinant of a Matrix18

2.1.1 Determinants of order 2 and order 318

2.1.2 Permutations and Number of Inversions21

2.1.3 Determinant of an n×n Matrix23

Exercises 2.126

2.2 Properties of Determinants26

Exercises 2.231

2.3 The Cofactor Expansion of a Determinant33

Exercises 2.338

2.4 Cramer's Rule39

Exercises 2.442

Chapter 3 Matrices Algebra43

3.1 Matrices Arithmetic43

3.1.1 Matrices43

3.1.2 Matrix Addition44

3.1.3 Scalar Multiplication45

3.1.4 Matrix Multiplication46

3.1.5 Powers of a Square Matrix48

Exercise 3.149

3.2 Special Matrices50

3.2.1 The Identity Matrix50

3.2.2 The Diagonal Matrix51

3.2.3 Triangular Matrices51

3.2.4 The Transpose of a Matrix52

3.2.5 Symmetric and Skew-Symmetric Matrices54

3.2.6 The Determinant of the Product AB54

3.2.7 Adjoint of a Matrix55

Exercises 3.256

3.3 The Inverse of a Matrix57

3.3.1 The Inverse of a Matrix57

3.3.2 Properties of Invertible Matrices61

Exercises 3.362

3.4 Partitioned Matrices63

3.4.1 Partitioned Matrices63

3.4.2 Operations on Partitioned Matrices64

Exercises 3.471

3.5 Elementary Matrices72

3.5.1 Elementary Matrices72

3.5.2 Theorems of Invertible Matrices76

3.5.3 Computing the Inverse by Elementary Row Operations77

3.5.4 Solving Matrix Equations by Elementary Row Operations78

Exercises 3.580

3.6 The Rank of a Matrix82

3.6.1 The Rank of a Matrix82

3.6.2 Properties of the Ranks of Matrices86

Exercises 3.689

Chapter 4 Structure of Solutions for Systems of Linear Equations91

4.1 Existence and Uniqueness of Solutions91

Exercises 4.195

4.2 Vectors and Operations on Vectors96

Exercises 4.298

4.3 Linear Relation Among Vectors98

4.3.1 Linear Combination of Vectors98

4.3.2 Linear Dependency100

Exercises 4.3105

4.4 Rank of a Vector Set107

4.4.1 Equivalence Relation Between Two Vector Sets107

4.4.2 Rank of a Vector Set109

4.4.3 Relation Between Rank of a Matrix and Rank of a Vector Set110

Exercises 4.4113

4.5 Structure of Solutions for Systems of Linear Equations113

4.5.1 Homogeneous Linear Systems114

4.5.2 Nonhomogeneous Linear Systems120

Exercises 4.5124

4.6 Vector Spaces126

4.6.1 Vector Spaces,Subspaces and Spanning Sets126

4.6.2 Basis and Coordinate Vector129

Exercises 4.6132

Chapter 5 Similar Matrices and Quadratic Forms134

5.1 Eigenvalues and Eigenvectors134

Exercises 5.1141

5.2 Similar Matrices142

5.2.1 Definition of Similarity142

5.2.2 Diagonalization143

Exercises 5.2147

5.3 Diagonalization of Symmetric Matrices148

5.3.1 Inner Product,Length,and Orthogonality148

5.3.2 Gram-Schmidt Process150

5.3.3 Orthogonal Matrices and Orthogonal Transformation151

5.3.4 Diagonalization of Symmetric Matrices152

Exercises 5.3155

5.4 Quadratic Forms156

5.4.1 Quadratic Forms156

5.4.2 Diagonalizing a Quadratic Form158

5.4.3 Definiteness of a Quadratic Form160

Exercises 5.4161

Answers to Exercises163

References175