内容简介

随着计算机技术的发展，线性代数课程的重要性越来越突出。同时，现代软件已经为显著改进授课方式提供了可能。本书作者多年讲授线性代数课程，并在教学过程中不断探索更利于学生理解的新教学方法，从而使本书更加适合作为线性代数课程的教材。

在第8版中，扩充了矩阵代数的知识，新增了向量积、实舒尔分解的内容，并增加了130多道新练习。

本书主要特点

 理论与应用有机结合。大量的实际应用贯穿于理论讲解的始终，体现了线性代数在各个领域中的广泛应用。

 示例丰富。便于读者理解相关的定义及原理，增强了读者学习的兴趣。

 习题安排错落有致。每一节的后面给出大量的习题，各章后面还有测试题，使学生有更多的演练机会，达到触类旁通的效果。

 紧密结合数学工具MATLAB。每章的后面都有基于MATLAB的上机练习，并在附录中介绍了MATLAB的基本用法。

作品目录

Contents

Preface iii

Matrices and Systems of Equations 1

1.1 Systems of Linear Equations 1

1.2 Row EchelonForm 11

1.3 Matrix Arithmetic 27

1.4 Matrix Algebra 44

1.5 Elementary Matrices 58

1.6 Partitioned Matrices 68

MATLAB R Exercises 77

ChapterTestA 81

ChapterTestB 82

2 Determinants 84

2.1 The Determinant of a Matrix 84

2.2 Properties of Determinants 91

2.3. AdditionalTopics and Applications 98

MATLAB Exercises 106

ChapterTestA 108

ChapterTestB 108

3 Vector Spaces 110

3.1 Definition and Examples 110

3.2 Subspaces 117

3.3 Linear Independence 127

3.4 Basis and Dimension 138

3.5 Change of Basis 144

3.6 Row Space and Column Space 154

MATLAB Exercises 162

ChapterTestA 164

ChapterTestB 164

4 Linear Transformations 166

4.1 Definition and Examples 166

4.2 Matrix Representationsof LinearTransformations 175

4.3 Similarity 189

MATLAB Exercises 195

ChapterTestA 196

ChapterTestB 197

5 Orthogonality 198

5.1 The Scalar Product in Rn 199

5.2 Orthogonal Subspaces 214

5.3 Least Squares Problems 222

5.4 Inner Product Spaces 232

5.5 Orthonormal Sets 241

5.6 The Gram–Schmidt Orthogonalization Process 259

5.7 Orthogonal Polynomials 269

MATLAB Exercises 277

ChapterTestA 279

ChapterTestB 280

6 Eigenvalues 282

6.1 Eigenvalues and Eigenvectors 283

6.2 Systems of Linear Differential Equations 296

6.3 Diagonalization 307

6.4 Hermitian Matrices 324

6.5 The SingularValue Decomposition 337

6.6 QuadraticForms 351

6.7 Positive Definite Matrices 364

6.8 Nonnegative Matrices 372

MATLAB Exercises 378

ChapterTestA 384

ChapterTestB 384

7 Numerical Linear Algebra 386

7.1 Floating-Point Numbers 387

7.2 Gaussian Elimination 391

7.3 Pivoting Strategies 398

7.4 Matrix Norms and Condition Numbers 403

7.5 Orthogonal Transformations 417

7.6 The Eigenvalue Problem 428

7.7 Least Squares Problems 437

MATLAB Exercises 448

Chapter Test A 454

Chapter Test B 454

8 Iterative Methods Web.

9 Canonical Forms Web.

Appendix: MATLAB 456

Bibliography 468

Answers to Selected Exercises 471

Index 485

. Optional sections. These sections are not prerequisites for any other sections of the book.

. Web: The supplemental Chapters 8 and 9 can be downloaded from the internet. See the section of the Preface on supplemental Web materials.