线性代数及其应用（第三版）（英文版）

本书是对线性代数及其有趣应用的基本介绍。在前两版的基础上，第三版提供了更多的形象化概念、应用（如1.6节中的列昂捷夫经济学模型、化学方程组和业务流），以及增强的Web支持。

线性代数是处理矩阵和向量空间的数学分支科学，在现代数学的各个领域都有应用。本书主要包括线性方程组、矩阵代数、行列式、向量空间、特征值和特征向量、正交性和*小二乘方、对称矩阵和二次型等内容。本书的目的是使学生掌握线性代数*基本的概念、理论和证明。首先以常见的方式，具体介绍了线性独立、子空间、向量空间和线性变换等概念，然后逐渐展开，*后在抽象地讨论概念时，它们就变得容易理解多了。

David C. Lay：美国奥罗拉大学学士，加州大学洛杉矶分校硕士、博士，教育家。1976年起开始在马里兰大学从事数学教学与研究工作，阿姆斯特丹大学、自由大学、德国凯撒斯劳滕工业大学访问学者，在函数分析和线性代数领域发表文章30余篇。美国国家科学基金会资助的线性代数课程研究小组的创始人，参与编写了《函数分析、积分及其应用导论》和《线性代数精粹》等书。

CHAPTER 1 Linear Equations in Linear Algebra 1
Introductory Example： Linear Models in Economics and Engineering 1
1．1 Systems of Linear Equations 2
1．2 Row Reduction and Echelon Forms 14
1．3 Vector Equations 28
1．4 The Matrix Equation Ax = b 40
1．5 Solution Sets of Linear Systems 50
1．6 Applications of Linear Systems 57
1．7 Linear Independence 65
1．8 Introduction to Linear Transformations 73
1．9 The Matrix of a Linear Transformations 82
1．10 Linear Models in Business， Science， and Engineering 92
Supplementary Exercises 102

CHAPTER 2 Matrix Algebra 105
Introductory Example： Computer Models in Aircraft Design 105
2．1 Matrix Operations 107
2．2 The Inverse of a Matrix 118
2．3 Characterizations of Invertible Matrices 128
2．4 Partioned Matrices 134
2．5 Matrix Factorizations 142
2．6 The Leontief Input-Output Modes 152
2．7 Applications to Computer Graphics 158
2．8 Subspaces of Rn 167
2．9 Dimension and Rank 176
Supplementary Exercises 183

CHAPTER 3 Determinants 185
Introductory Example： Determinants in Analytic Geometry 185
3．1 Introduction to Determinants 186
3．2 Properties of Determinants 192
3．3 Cramer’s Rule， Volume， and Linear Transformations 201
Supplementary Exercises 211

CHAPTER 4 Vector Spaces 215
Introductory Example： Space Flight and Control Systems 215
4．1 Vector Spaces and Subspaces 216
4．2 Null Space， Column Spaces， and Linear Transformations 226
4．3 Linearly Independent Sets： Bases 237
4．4 Coordinate Systems 246
4．5 The Dimension of a Vector Space 256
4．6 Rank 262
4．7 Change of Basis 271
4．8 Applications to Difference Equations 277
4．9 Applications to Markov Chains 288
Supplementary Exercises 299

CHAPTER 5 Eigenvalues and Eigenvectors 301
Introductory Example： Dynamical Systems and Spotted Owls 301
5．1 Eigenvectors and Eignevalues 302
5．2 The Characteristic Equation 310
5．3 Diagonalization 319
5．4 Eigenvectors and Linear Transformations 327
5．5 Complex Eigenvalues 335
5．6 Discrete Dynamical Systems 342
5．7 Applications to Differential Equations 353
5．8 Iterative Estimates for Eigenvalues 363
Supplementary Exercises 370

CHAPTER 6 Orthogonality and Least Squares 373
Introductory Example： Readjusting the North American Datum 373
6．1 Inner Product， Length， and Orthogonality 375
6．2 Orthogonal Sets 384
6．3 Orthogonal Projections 394
6．4 The Gram-Schmidt Process 402
6．5 Least-Squares Problems 409
6．6 Applications to Linear Models 419
6．7 Inner Product Spaces 427
6．8 Applications of Inner Product Spaces 436
Supplementary Exercises 444

CHAPTER 7 Symmetric Matrices and Quadratic Forms 447
Introductory Example： Multichannel Image Processing 447
7．1 Diagonalization of Symmetric Matices 449
7．2 Quadratic Forms 455
7．3 Constrained Optimization 463
7．4 The Singular Value Decomposition 471
7．5 Applications to Image Processing and Statistics 482
Supplementary Exercises 444

Appendixes
A Uniqueness of the Reduced Echelon Form A1
B Complex Numbers A3

Glossary A9
Answers to Odd-Numbered Exercises A19
Index I1