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Chapter 1 Systems of Linear Equations and Elementary Operations on Matrices 1

1.1 Systems of Linear Equations 1

1.1.1 Definition 1

1.1.2 Equivalent Systems and Gaussian Elimination Method 3

Exercises 1.1 8

1.2 Elementary Operations on Matrices 9

1.2.1 Elementary Row Operations 9

1.2.2 Row Echelon Forms 11

1.2.3 Standard Form of a Matrix 15

Exercises 1.2 15

Chapter 2 Determinants 18

2.1 The Determinant of a Matrix 18

2.1.1 Determinants of order 2 and order 3 18

2.1.2 Permutations and Number of Inversions 21

2.1.3 Determinant of an n×n Matrix 23

Exercises 2.1 26

2.2 Properties of Determinants 26

Exercises 2.2 31

2.3 The Cofactor Expansion of a Determinant 33

Exercises 2.3 38

2.4 Cramer's Rule 39

Exercises 2.4 42

Chapter 3 Matrices Algebra 43

3.1 Matrices Arithmetic 43

3.1.1 Matrices 43

3.1.2 Matrix Addition 44

3.1.3 Scalar Multiplication 45

3.1.4 Matrix Multiplication 46

3.1.5 Powers of a Square Matrix 48

Exercise 3.1 49

3.2 Special Matrices 50

3.2.1 The Identity Matrix 50

3.2.2 The Diagonal Matrix 51

3.2.3 Triangular Matrices 51

3.2.4 The Transpose of a Matrix 52

3.2.5 Symmetric and Skew-Symmetric Matrices 54

3.2.6 The Determinant of the Product AB 54

3.2.7 Adjoint of a Matrix 55

Exercises 3.2 56

3.3 The Inverse of a Matrix 57

3.3.1 The Inverse of a Matrix 57

3.3.2 Properties of Invertible Matrices 61

Exercises 3.3 62

3.4 Partitioned Matrices 63

3.4.1 Partitioned Matrices 63

3.4.2 Operations on Partitioned Matrices 64

Exercises 3.4 71

3.5 Elementary Matrices 72

3.5.1 Elementary Matrices 72

3.5.2 Theorems of Invertible Matrices 76

3.5.3 Computing the Inverse by Elementary Row Operations 77

3.5.4 Solving Matrix Equations by Elementary Row Operations 78

Exercises 3.5 80

3.6 The Rank of a Matrix 82

3.6.1 The Rank of a Matrix 82

3.6.2 Properties of the Ranks of Matrices 86

Exercises 3.6 89

Chapter 4 Structure of Solutions for Systems of Linear Equations 91

4.1 Existence and Uniqueness of Solutions 91

Exercises 4.1 95

4.2 Vectors and Operations on Vectors 96

Exercises 4.2 98

4.3 Linear Relation Among Vectors 98

4.3.1 Linear Combination of Vectors 98

4.3.2 Linear Dependency 100

Exercises 4.3 105

4.4 Rank of a Vector Set 107

4.4.1 Equivalence Relation Between Two Vector Sets 107

4.4.2 Rank of a Vector Set 109

4.4.3 Relation Between Rank of a Matrix and Rank of a Vector Set 110

Exercises 4.4 113

4.5 Structure of Solutions for Systems of Linear Equations 113

4.5.1 Homogeneous Linear Systems 114

4.5.2 Nonhomogeneous Linear Systems 120

Exercises 4.5 124

4.6 Vector Spaces 126

4.6.1 Vector Spaces,Subspaces and Spanning Sets 126

4.6.2 Basis and Coordinate Vector 129

Exercises 4.6 132

Chapter 5 Similar Matrices and Quadratic Forms 134

5.1 Eigenvalues and Eigenvectors 134

Exercises 5.1 141

5.2 Similar Matrices 142

5.2.1 Definition of Similarity 142

5.2.2 Diagonalization 143

Exercises 5.2 147

5.3 Diagonalization of Symmetric Matrices 148

5.3.1 Inner Product,Length,and Orthogonality 148

5.3.2 Gram-Schmidt Process 150

5.3.3 Orthogonal Matrices and Orthogonal Transformation 151

5.3.4 Diagonalization of Symmetric Matrices 152

Exercises 5.3 155

5.4 Quadratic Forms 156

5.4.1 Quadratic Forms 156

5.4.2 Diagonalizing a Quadratic Form 158

5.4.3 Definiteness of a Quadratic Form 160

Exercises 5.4 161

Answers to Exercises 163

References 175