綫性代數是處理矩陣和嚮量空間的數學分支科學，在現代數學的各個領域都有應用。本書主要包括綫性方程組、矩陣代數、行列式、嚮量空間、特徵值和特徵嚮量、正交性和*小二乘方、對稱矩陣和二次型等內容。本書的目的是使學生掌握綫性代數*基本的概念、理論和證明。首先以常見的方式，具體介紹瞭綫性獨立、子空間、嚮量空間和綫性變換等概念，然後逐漸展開，*後在抽象地討論概念時，它們就變得容易理解多瞭。

作者簡介

David C. Lay：美國奧羅拉大學學士，加州大學洛杉磯分校碩士、博士，教育傢。1976年起開始在馬裏蘭大學從事數學教學與研究工作，阿姆斯特丹大學、自由大學、德國凱撒斯勞滕工業大學訪問學者，在函數分析和綫性代數領域發錶文章30餘篇。美國國傢科學基金會資助的綫性代數課程研究小組的創始人，參與編寫瞭《函數分析、積分及其應用導論》和《綫性代數精粹》等書。

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