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Chapter 1 Equation Systems and Matrices 1
1.1 Systems of Linear Equations 1
1.1.1 Brief History of Algebra and Linear Algebra 1
1.1.2 Systems of Linear Equations 2
1.1.3 Strict Triangular Form of Linear Systems 7
1.2 Linear System in Matrix 10
1.2.1 Matrix Notations 10
1.2.2 Solving Linear Systems 11
1.3 Reduced Row Echelon Form 15
1.3.1 Row Echelon Form 15
1.3.2 Gauss Elimination 18
1.3.3 Reduced Row Echelon 21
1.4 Consistency of Linear Systems 23
1.4.1 Overdetermined Systems 23
1.4.2 Underdetermined Systems 27
1.4.3 Homogeneous Systems 28
Chapter 2 Matrix Algebra 30
2.1 Notations and Operations 30
2.1.1 Matrix Notations 30
2.1.2 Matrix Operations 31
2.1.3 Algebraic Rules of Matrix Operations 34
2.2 Inverse and Transpose of Matrices 38
2.2.1 Identity Matrix 38
2.2.2 Matrix Inverse 39
2.2.3 The Transpose of a Matrix 41
2.2.4 Triangular and Diagonal Matrices 42
2.3 Partitioned Matrices 44
2.3.1 The Notations of Partitioned Matrices 44
2.3.2 Block Addition and Scalar Multiplication 46
2.3.3 Block Multiplication 47
2.4 Linear Combination of Vectors 52
2.4.1 Linear Combination of Vectors 52
2.4.2 Equivalent Systems 53
2.4.3 Elementary Matrices 55
2.4.4 Find the Inverse Matrix.59
2.5 The Determinant of a Matrix 61
2.5.1 CASE ¢ñ The Determinant of 1 ¡ê 1 Matrices 62
2.5.2 CASE ¢ò The Determinant of 2 ¡ê 2 Matrices 62
2.5.3 CASE ¢ó 3 ¡ê 3 Matrices 63
2.5.4 CASE ¢ô The Determinant of n ¡ê n Matrices 64
2.6 Properties of Determinants 67
2.6.1 Determinant of the Transposed Matrix 67
2.6.2 Determinant of Triangular Matrices.68
2.6.3 Determinant of Matrices with All Zeros in a Row or Column 68
2.6.4 Determinant of Matrices with Identical Rows or Columns 69
2.6.5 ¤Laplace's De¡¥nition of Determinant by Using Subdeterminant 70
2.6.6 Algebraic Rules of Determinants 71
2.6.7 Determinant and Singularity of a Matrix 83
2.7 Cramer's Rule 85
2.7.1 The Adjoint of a Matrix 85
2.7.2 Cramer's Rule 86
Chapter 3 Vector Spaces 89
3.1 De¡¥nitions and Examples 89
3.1.1 Definitions 89
3.1.2 Examples 90
3.1.3 Euclidean Vector Space 91
3.1.4 Inner Product and Outer Product Expansion of Vectors 92
3.2 Subspaces 94
3.2.1 Definitions 94
3.2.2 The Null Space of a Matrix 95
3.2.3 The Span of Vectors 96
3.3 Linear Independence 102
3.3.1 Concepts and Examples 102
3.3.2 The Minimal Spanning Set of a Vector Space Matrix.109
3.4 Basis and Dimension 112
3.4.1 Basis of Vector Spaces 112
3.4.2 Dimension of Vector Spaces 114
3.5 Changing of Basis 117
3.5.1 Coordinate of Vector 117
3.5.2 Changing of Basis in R2 120
3.5.3 Changing of Basis in an n-dimensional Vector Space 122
3.6 Row Space and Column Space of Matrices 124
3.6.1 Concepts and Examples124
3.6.2 Rank of a Matrix 126
3.6.3 The Rank and Nullity Theorem 128
Chapter 4 Analytic Geometry 131
4.1 Analytic Geometry and Cartesian Coordinate System 131
4.1.1 Cartesian Coordinate System on Plane 133
4.1.2 Cartesian Coordinate System in Space 134
4.1.3 Vectors in Cartesian Coordinate System 135
4.2 Algebra in Euclidean Geometry 136
4.2.1 Euclidean Length 136
4.2.2 Included Angle of Two Vectors139
4.2.3 The Geometric Interpretations of Operations on Vectors 140
4.2.4 The Projection of Vectors142
4.2.5 Inner Product 144
4.2.6 Cross Product 151
4.2.7 The Triple Scalar or Box Product 153
4.3 Planes and Lines 156
4.3.1 The Equation and Figure of Space Surface 157
4.3.2 The Equation of a Plane 158
4.3.3 The Relative Positions of Planes 162
4.3.4 The Equation of a Line 165
4.3.5 The Relative Positions of Lines 170
4.3.6 The Relative Positions Between a Line and a Plane 175
4.3.7 The Distance from a Point to a Plane or a Line 177
Chapter 5 Linear Transformation 180
5.1 Definition and Examples 180
5.2 The Image and Kernel 185
5.3 Matrix Representation of Linear Transformations 188
5.4 Similar Matrices 192
Chapter 6 Matrix Diagonalization 198
6.1 Inner Product and Inner Product Space 198
6.2 Orthonormal Sets and Orthogonal Subspaces 204
6.2.1 Orthonormal Sets 204
6.2.2 Orthogonal Matrices 206
6.2.3 Orthogonal Subspaces 207
6.3 The Gram-Schmidt Orthogonalization Process 210
6.4 Eigenvalues and Eigenvectors 215
6.4.1 Concepts and Examples 216
6.4.2 The Product and Sum of the Eigenvalues 218
6.4.3 The Eigenvalues and Eigenvectors of Similar Matrices 219
6.5 Diagonalization 221
Chapter 7 Quadratic Form and Its Applications 226
7.1 Quadratic Form and Its Matrix Representation 226
7.2 The Diagonalization of Real Symmetric Matrices 231
7.3 Conic Sections and Quadric Surfaces 234
7.3.1 Conic Sections 235
7.3.2 Quadric Surfaces 244
Bibliography 250