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An Introduction to Linear Algebra

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Table of Contents
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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An Introduction to Linear Algebra
$15.72