# Elementary Linear Algebra (Classic Version), 2nd edition

Published by Pearson (March 20, 2017) © 2018

**Lawrence E. Spence**Illinois State University**Arnold J. Insel**Illinois State University**Stephen H. Friedberg**Illinois State University

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For sophomore-level courses in Linear Algebra.

### A modern classic

Based on the recommendations of the Linear Algebra Curriculum Study Group, **Elementary Linear Algebra** offers a matrix-oriented approach with more emphasis on problem solving and applications. Throughout the **2nd Edition,** use of technology is encouraged. The focus is on matrix arithmetic, systems of linear equations, properties of Euclidean n-space, eigenvalues and eigenvectors, and orthogonality. Although matrix-oriented, the text provides a solid coverage of vector spaces.

This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price.

### Hallmark features of this title

- Examples are accompanied by similar practice problems that enable students to test their understanding of the material Complete solutions to the practice problems are included with the text
- All computational exercises are designed so that the calculations involve “nice” numbers.
- Most sections include approximately 20 true/false exercises designed to test students' understanding of the conceptual ideas in each section. Answers to every true/false are given in the book.
- Each chapter ends with a set of review exercises that provide practice with all the main topics of each chapter.
- For a proof-oriented course, the authors have included a significant number of accessible exercises requiring proofs. They are ordered according to difficulty.
- A wide variety of applications include traffic flow, scheduling, and Google searches.

### New and updated features of this title

- Significantly increased number of computation exercises.
- Complete redesign and development of the text.

**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**

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