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#### An Introduction to Linear Algebra

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Contents
Chapter 1　Linear Systems and Matrices　1
1.1　Introduction to Linear Systems and Matrices　1
1.1.1　Linear equations and linear systems　1
1.1.2　Matrices　3
1.1.3　Elementary row operations　4
1.2　Gauss-Jordan Elimination　5
1.2.1　Reduced row-echelon form　5
1.2.2　Gauss-Jordan elimination　6
1.2.3　Homogeneous linear systems　9
1.3　Matrix Operations　11
1.3.1　Operations on matrices　11
1.3.2　Partition of matrices　13
1.3.3　Matrix product by columns and by rows　13
1.3.4　Matrix product of partitioned matrices　14
1.3.5　Matrix form of a linear system　15
1.3.6　Transpose and trace of a matrix　16
1.4　Rules of Matrix Operations and Inverses　18
1.4.1　Basic properties of matrix operations　19
1.4.2　Identity matrix and zero matrix　20
1.4.3　Inverse of a matrix　21
1.4.4　Powers of a matrix　23
1.5　 Elementary Matrices and a Method for Finding A.1　24
1.5.1　Elementary matrices and their properties　24
1.5.2　Main theorem of invertibility　26
1.5.3　A method for finding A.1　27
1.6　Further Results on Systems and Invertibility　28
1.6.1　A basic theorem　28
1.6.2　Properties of invertible matrices　29
1.7　Some Special Matrices　31
1.7.1　Diagonal and triangular matrices　32
1.7.2　Symmetric matrix　34
Exercises　35
Chapter 2　Determinants　42
2.1　Determinant Function　42
2.1.1　Permutation， inversion， and elementary product　42
2.1.2　Definition of determinant function　44
2.2　Evaluation of Determinants　44
2.2.1　Elementary theorems　44
2.2.2　A method for evaluating determinants　46
2.3　Properties of Determinants　46
2.3.1　Basic properties　47
2.3.2　Determinant of a matrix product　48
2.3.3　Summary　50
2.4　Cofactor Expansions and Cramer’s Rule　51
2.4.1　Cofactors　51
2.4.2　Cofactor expansions　51
2.4.3　Adjoint of a matrix　53
2.4.4　Cramer’s rule　54
Exercises　55
Chapter 3 Euclidean Vector Spaces　61
3.1　Euclidean n-Space　61
3.1.1　n-vector space　61
3.1.2　Euclidean n-space　62
3.1.3　Norm， distance， angle， and orthogonality　63
3.1.4　Some remarks　65
3.2　Linear Transformations from Rn to Rm　66
3.2.1　Linear transformations from Rn to Rm　66
3.2.2　Some important linear transformations　67
3.2.3　Compositions of linear transformations　69
3.3　Properties of Transformations　70
3.3.1　Linearity conditions　70
3.3.2　Example　71
3.3.3　One-to-one transformations　72
3.3.4　Summary　73
Exercises　74
Chapter 4　General Vector Spaces　79
4.1　Real Vector Spaces　79
4.1.1　Vector space axioms　79
4.1.2　Some properties　81
4.2　Subspaces　81
4.2.1　Definition of subspace　82
4.2.2　Linear combinations　83
4.3　Linear Independence　85
4.3.1　Linear independence and linear dependence　86
4.3.2 　Some theorems　87
4.4　Basis and Dimension　88
4.4.1　Basis for vector space　88
4.4.2　Coordinates　89
4.4.3　Dimension　91
4.4.4　Some fundamental theorems　93
4.4.5　Dimension theorem for subspaces　95
4.5　Row Space， Column Space， and Nullspace　97
4.5.1　Definition of row space， column space， and nullspace　97
4.5.2　Relation between solutions of Ax = 0 and Ax=b　98
4.5.3　Bases for three spaces　100
4.5.4　A procedure for finding a basis for span(S)　102
4.6　Rank and Nullity　103
4.6.1　Rank and nullity　104
4.6.2　Rank for matrix operations　106
4.6.3　Consistency theorems　107
4.6.4　Summary　109
Exercises　110
Chapter 5　Inner Product Spaces　115
5.1　Inner Products　115
5.1.1　General inner products　115
5.1.2　Examples　116
5.2　Angle and Orthogonality　119
5.2.1　Angle between two vectors and orthogonality　119
5.2.2　Properties of length， distance， and orthogonality　120
5.2.3　Complement　121
5.3　Orthogonal Bases and Gram-Schmidt Process　122
5.3.1　Orthogonal and orthonormal bases　122
5.3.2　Projection theorem　125
5.3.3　Gram-Schmidt process　128
5.3.4　QR-decomposition　130
5.4　Best Approximation and Least Squares　133
5.4.1　Orthogonal projections viewed as approximations　134
5.4.2　Least squares solutions of linear systems　135
5.4.3　Uniqueness of least squares solutions　136
5.5　Orthogonal Matrices and Change of Basis.　138
5.5.1　Orthogonal matrices　138
5.5.2　Change of basis　140
Exercises　144
Chapter 6　Eigenvalues and Eigenvectors　149
6.1　Eigenvalues and Eigenvectors　149
6.1.1　Introduction to eigenvalues and eigenvectors　149
6.1.2　Two theorems concerned with eigenvalues　150
6.1.3　Bases for eigenspaces　151
6.2　Diagonalization　152
6.2.1　Diagonalization problem　152
6.2.2　Procedure for diagonalization　153
6.2.3　Two theorems concerned with diagonalization　155
6.3　Orthogonal Diagonalization　156
6.4　Jordan Decomposition Theorem　160
Exercises　162
Chapter 7　Linear Transformations　166
7.1　General Linear Transformations　166
7.1.1　Introduction to linear transformations　166
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