**PREFACE ****ix**

**TO THE STUDENT** **xv**

**CHAPTER 1 MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS ****1**

1.1 Matrices and Vectors **1**

1.2 Linear Combinations, Matrix—Vector Products, and Special Matrices **11**

1.3 Systems of Linear Equations **25**

1.4 Gaussian Elimination **39**

1.5∗ Applications of Systems of Linear Equations **54**

1.6 The Span of a Set of Vectors **64**

1.7 Linear Dependence and Linear Independence **73**

Chapter 1 Review Exercises

Chapter 1 MATLAB Exercises

**CHAPTER 2 MATRICES AND LINEAR TRANSFORMATIONS** **90**

2.1 Matrix Multiplication **90**

2.2∗ Applications of Matrix Multiplication **101**

2.3 Invertibility and Elementary Matrices **117**

2.4 The Inverse of a Matrix **130**

2.5∗ Partitioned Matrices and Block Multiplication **141**

2.6∗ The *LU *Decomposition of a Matrix **147**

2.7 Linear Transformations and Matrices **162**

2.8 Composition and Invertibility of Linear Transformations **175**

Chapter 2 Review Exercises

Chapter 2 MATLAB Exercises

**CHAPTER 3 DETERMINANTS** **192**

3.1 Cofactor Expansion **192**

3.2 Properties of Determinants **204**

Chapter 3 Review Exercises

Chapter 3 MATLAB Exercises

**CHAPTER 4 SUBSPACES AND THEIR PROPERTIES** **218**

4.1 Subspaces **218**

4.2 Basis and Dimension **232**

4.3 The Dimension of Subspaces Associated with a Matrix **245**

4.4 Coordinate Systems **254**

4.5 Matrix Representations of Linear Operators **266**

Chapter 4 Review Exercises

Chapter 4 MATLAB Exercises

**CHAPTER 5 EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION ****282**

5.1 Eigenvalues and Eigenvectors **282**

5.2 The Characteristic Polynomial **291**

5.3 Diagonalization of Matrices **302**

5.4∗ Diagonalization of Linear Operators **314**

5.5∗ Applications of Eigenvalues **323**

Chapter 5 Review Exercises

Chapter 5 MATLAB Exercises

**CHAPTER 6 ORTHOGONALITY ****347**

6.1 The Geometry of Vectors **347**

6.2 Orthogonal Vectors **360**

6.3 Orthogonal Projections **374**

6.4 Least-Squares Approximations and Orthogonal Projections **388**

6.5 Orthogonal Matrices and Operators **398**

6.6 Symmetric Matrices **412**

6.7∗ Singular Value Decomposition **425**

6.8∗ Principal Component Analysis **443**

6.9∗ Rotations of *R*3 and Computer Graphics **452**

Chapter 6 Review Exercises

Chapter 6 MATLAB Exercises

**CHAPTER 7 VECTOR SPACES ****473**

7.1 Vector Spaces and Their Subspaces **473**

7.2 Linear Transformations **485**

7.3 Basis and Dimension **495**

7.4 Matrix Representations of Linear Operators **505**

**7.5 **Inner Product Spaces **517**

Chapter 7 Review Exercises

Chapter 7 MATLAB Exercises

**APPENDICES ****534**

A Sets **534**

B Functions **536**

C Complex Numbers **539**

D Matlab **544**

E The Uniqueness of the Reduced Row Echelon Form **558**

REFERENCES **561**

ANSWERS TO SELECTED EXERCISES **562**

LIST OF FREQUENTLY USED SYMBOLS **608**

INDEX **609**