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內容介紹

編輯推薦

介紹瞭綫性代數最基本的概念、理論和證明。包含瞭大量與實際問題相關的習題，並附有習題答案。提供瞭豐富的應用以解釋工程學、計算機科學、數學、物理學、生物學、經濟學和統計學中的基本原理及簡單計算。提齣瞭矩陣-嚮量乘法的動態和圖形觀點，將嚮量空間的概念引入綫性係統的學習中，介紹瞭正交性和最小二乘方問題。強調瞭在科學和工程學領域，計算機對綫性代數發展和實踐的影響。用小圖標標記的部分可在網站www.laylinalgebra.com或www.mymathlab.com上找到相應的技術支持，包含習題的數據文件、實例學習和應用方案等內容。

 

 

內容簡介

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綫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，最後在抽象地討論概念時，它們就變得容易理解多瞭。

 

 

目    錄
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

 

 

作者介紹
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。自1976年起開始於馬裏蘭大學從事數學的教學與研究工作，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。

關聯推薦
本書是介紹性的綫性代數教材，內容翔實，層次清晰，適閤作為高等院校理工科數學課的雙語教學用書，也可作為公司職員及工程學研究人員的參考書。
目錄
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

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