線性代數是處理矩陣和向量空間的數學分支科學，在現代數學的各個領域都有應用。本書主要包括線性方程組、矩陣代數、行列式、向量空間、特征值和特征向量、正交性和最小二乘方、對稱矩陣和二次型等內容。本書的目的是使學生掌握線性代數最基本的概念、理論和證明。首先以常見的方式，具體介紹了線性獨立、子空間、向量空間和線性變換等概念，然后逐漸展開，最后在抽象地討論概念時，它們就變得容易理解多了。
本書是對線性代數及其有趣應用的基本介紹。在前兩版的基礎上，第三版提供了更多的形象化概念、應用（如1.6節中的列昂捷夫經濟學模型、化學方程組和業務流），以及增強的Web支持。
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，教育家。1976年起開始在馬里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，在函數分析和線性代數領域發表文章30余篇。美國國家科學基金會資助的線性代數課程研究小組的創始人，參與編寫了《函數分析、積分及其應用導論》和《線性代數精粹》等書。
CHAPTER 1 Linear Equations in Linear Algebra 1Introductory Example: Linear Models in Economics and Engineering 11.1Systems of Linear Equations 21.2Row Reduction and Echelon Forms 141.3Vector Equations 281.4The Matrix Equation Ax = b 401.5Solution Sets of Linear Systems 501.6Applications of Linear Systems 571.7Linear Independence 651.8Introduction to Linear Transformations 731.9The Matrix of a Linear Transformations 821.10Linear Models in Business, Science, and Engineering 92Supplementary Exercises 102CHAPTER 2 Matrix Algebra 105Introductory Example: Computer Models in Aircraft Design 1052.1Matrix Operations 1072.2The Inverse of a Matrix 1182.3Characterizations of Invertible Matrices 1282.4Partioned Matrices 1342.5Matrix Factorizations 1422.6The Leontief Input-Output Modes 1522.7Applications to Computer Graphics 1582.8Subspaces of Rn 1672.9Dimension and Rank 176Supplementary Exercises 183CHAPTER 3 Determinants 185Introductory Example: Determinants in Analytic Geometry 1853.1Introduction to Determinants 1863.2Properties of Determinants 1923.3Cramer’s Rule, Volume, and Linear Transformations 201Supplementary Exercises 211CHAPTER 4 Vector Spaces 215Introductory Example: Space Flight and Control Systems 2154.1Vector Spaces and Subspaces 2164.2Null Space, Column Spaces, and Linear Transformations 2264.3Linearly Independent Sets: Bases 2374.4Coordinate Systems 2464.5The Dimension of a Vector Space 2564.6Rank 2624.7Change of Basis 2714.8Applications to Difference Equations 2774.9Applications to Markov Chains 288Supplementary Exercises 299CHAPTER 5 Eigenvalues and Eigenvectors 301Introductory Example: Dynamical Systems and Spotted Owls 3015.1Eigenvectors and Eignevalues 3025.2The Characteristic Equation 3105.3Diagonalization 3195.4Eigenvectors and Linear Transformations 3275.5Complex Eigenvalues 3355.6Discrete Dynamical Systems 3425.7Applications to Differential Equations 3535.8Iterative Estimates for Eigenvalues 363Supplementary Exercises 370CHAPTER 6 Orthogonality and Least Squares 373Introductory Example: Readjusting the North American Datum 3736.1Inner Product, Length, and Orthogonality 3756.2Orthogonal Sets 3846.3Orthogonal Projections 3946.4The Gram-Schmidt Process 4026.5Least-Squares Problems 4096.6Applications to Linear Models 4196.7Inner Product Spaces 4276.8Applications of Inner Product Spaces 436Supplementary Exercises 444CHAPTER 7 Symmetric Matrices and Quadratic Forms 447Introductory Example: Multichannel Image Processing 4477.1Diagonalization of Symmetric Matices 4497.2Quadratic Forms 4557.3Constrained Optimization 4637.4The Singular Value Decomposition 4717.5Applications to Image Processing and Statistics 482Supplementary Exercises 444AppendixesA Uniqueness of the Reduced Echelon Form A1B Complex Numbers A3Glossary A9Answers to Odd-Numbered Exercises A19Index I1
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