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An Introduction to Linear Algebra

An Introduction to Linear Algebra

书籍作者:金小庆 ISBN:9787030721631
书籍语言:简体中文 连载状态:全集
电子书格式:pdf,txt,epub,mobi,azw3 下载次数:1044
创建日期:2023-04-28 发布日期:2023-04-28
运行环境:PC/Windows/Linux/Mac/IOS/iPhone/iPad/Kindle/Android/安卓/平板
内容简介

线性代数是现代数学的基础并广泛应用于科学和工程领域中。随着计算机的发展,线性代数和计算科学紧密结合,它在互联网领域,如网络搜索、目标识别、视频图像处理等领域也产生了重大的影响:线性代数极其重要的应用之一就是Goolge的创建,超级复杂的排名算法就是在线性代数的辅助下创建的。线性代数是最富创造性的数学工具。

由澳门大学金小庆教授、刘璇、刘伟辉博士和杭州电子科技大学赵志副教授撰写的《线性代数导引》一书共八章,包含线性方程组、矩阵、向量空间、特征值和特征向量、线性变换等。在最后一章,作者从书中的基本概念出发,对最近提出的Böttcher-Wenzel 猜想给出了初等证明。附录还初步讨论了向量空间公理的独立性。本书每章后都附有一定数量难度适宜的习题。

拥有此书,您与学好线性代数只有一步之遥。


目录
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
7.1.