# Elementary Linear Algebra

## 5th Edition

**Authors:**Stephen Andrilli David Hecker

**Hardcover ISBN:**9780128008539

**eBook ISBN:**9780128010471

**Imprint:**Academic Press

**Published Date:**25th February 2016

**Page Count:**806

## Description

Elementary Linear Algebra, 5th edition, by Stephen Andrilli and David Hecker, is a textbook for a beginning course in linear algebra for sophomore or junior mathematics majors. This text provides a solid introduction to both the computational and theoretical aspects of linear algebra. The textbook covers many important real-world applications of linear algebra, including graph theory, circuit theory, Markov chains, elementary coding theory, least-squares polynomials and least-squares solutions for inconsistent systems, differential equations, computer graphics and quadratic forms. Also, many computational techniques in linear algebra are presented, including iterative methods for solving linear systems, **LDU** Decomposition, the Power Method for finding eigenvalues, **QR** Decomposition, and Singular Value Decomposition and its usefulness in digital imaging.

The most unique feature of the text is that students are nurtured in the art of creating mathematical proofs using linear algebra as the underlying context. The text contains a large number of worked out examples, as well as more than 970 exercises (with over 2600 total questions) to give students practice in both the computational aspects of the course and in developing their proof-writing abilities. Every section of the text ends with a series of true/false questions carefully designed to test the students’ understanding of the material. In addition, each of the first seven chapters concludes with a thorough set of review exercises and additional true/false questions. Supplements to the text include an Instructor’s Manual with answers to all of the exercises in the text, and a Student Solutions Manual with detailed answers to the starred exercises in the text. Finally, there are seven additional web sections available on the book’s website to instructors who adopt the text.

## Key Features

- Builds a foundation for math majors in reading and writing elementary mathematical proofs as part of their intellectual/professional development to assist in later math courses
- Presents each chapter as a self-contained and thoroughly explained modular unit.
- Provides clearly written and concisely explained ancillary materials, including four appendices expanding on the core concepts of elementary linear algebra
- Prepares students for future math courses by focusing on the conceptual and practical basics of proofs

## Readership

upper level undergrads in math, physical science and engineering

## Table of Contents

- Dedication
- Preface for the Instructor
- Philosophy of the Text
- Major Changes for the Fifth Edition
- Plans for Coverage
- Prerequisite Chart for Later Sections
- Acknowledgments

- Preface to the Student
- A Light-Hearted Look at Linear Algebra Terms
- Symbol Table
- Computational & Numerical Techniques, Applications
- Chapter 1: Vectors and Matrices
- Abstract
- 1.1 Fundamental Operations with Vectors
- 1.2 The Dot Product
- 1.3 An Introduction to Proof Techniques
- 1.4 Fundamental Operations with Matrices
- 1.5 Matrix Multiplication
- Review Exercises for Chapter 1

- Chapter 2: Systems of Linear Equations
- Abstract
- 2.1 Solving Linear Systems Using Gaussian Elimination
- 2.2 Gauss-Jordan Row Reduction and Reduced Row Echelon Form
- 2.3 Equivalent Systems, Rank, and Row Space
- 2.4 Inverses of Matrices
- Review Exercises for Chapter 2

- Chapter 3: Determinants and Eigenvalues
- Abstract
- 3.1 Introduction to Determinants
- 3.2 Determinants and Row Reduction
- 3.3 Further Properties of the Determinant
- 3.4 Eigenvalues and Diagonalization
- Review Exercises for Chapter 3

- Chapter 4: Finite Dimensional Vector Spaces
- Abstract
- 4.1 Introduction to Vector Spaces
- 4.2 Subspaces
- 4.3 Span
- 4.4 Linear Independence
- 4.5 Basis and Dimension
- 4.6 Constructing Special Bases
- 4.7 Coordinatization
- Review Exercises for Chapter 4

- Chapter 5: Linear Transformations
- Abstract
- 5.1 Introduction to Linear Transformations
- 5.2 The Matrix of a Linear Transformation
- 5.3 The Dimension Theorem
- 5.4 One-to-One and Onto Linear Transformations
- 5.5 Isomorphism
- 5.6 Diagonalization of Linear Operators
- Review Exercises for Chapter 5

- Chapter 6: Orthogonality
- Abstract
- 6.1 Orthogonal Bases and the Gram-Schmidt Process
- 6.2 Orthogonal Complements
- 6.3 Orthogonal Diagonalization

- Chapter 7: Complex Vector Spaces and General Inner Products
- Abstract
- 7.1 Complex
*n*-Vectors and Matrices - 7.2 Complex Eigenvalues and Complex Eigenvectors
- 7.3 Complex Vector Spaces
- 7.4 Orthogonality in
- 7.5 Inner Product Spaces
- Review Exercises for Chapter 7

- Chapter 8: Additional Applications
- Abstract
- 8.1 Graph Theory
- 8.2 Ohm’s Law
- 8.3 Least-Squares Polynomials
- 8.4 Markov Chains
- 8.5 Hill Substitution: An Introduction to Coding Theory
- 8.6 Rotation of Axes for Conic Sections
- 8.7 Computer Graphics
- 8.8 Differential Equations
- 8.9 Least-Squares Solutions for Inconsistent Systems
- 8.10 Quadratic Forms

- Chapter 9: Numerical Techniques
- Abstract
- 9.1 Numerical Techniques for Solving Systems
- 9.2 LDU Decomposition
- 9.3 The Power Method for Finding Eigenvalues
- 9.4 QR Factorization
- 9.5 Singular Value Decomposition

- Appendix A: Miscellaneous Proofs
- Proof of Theorem 1.16, Part (1)
- Proof of Theorem 2.6
- Proof of Theorem 2.10
- Proof of Theorem 3.3, Part (3), Case 2
- Proof of Theorem 5.29
- Proof of Theorem 6.19

- Appendix B: Functions
- Functions: Domain, Codomain, and Range
- One-to-One and Onto Functions
- Composition and Inverses of Functions
- New Vocabulary
- Highlights
- Exercises for Appendix B

- Appendix C: Complex Numbers
- New Vocabulary
- Highlights
- Exercises for Appendix C

- Appendix D: Elementary Matrices
- Prerequisite: Section 2.4, Inverses of Matrices
- Elementary Matrices
- Representing a Row Operation as Multiplication by an Elementary Matrix
- Inverses of Elementary Matrices
- Using Elementary Matrices to Show Row Equivalence
- Nonsingular Matrices Expressed as a Product of Elementary Matrices
- New Vocabulary
- Highlights
- Exercises for Appendix D

- Appendix E: Answers to Selected Exercises
- Section 1.1 (p. 1–19)
- Section 1.2 (p. 19–34)
- Section 1.3 (p. 34–52)
- Section 1.4 (p. 52–65)
- Section 1.5 (p. 65–81)
- Chapter 1 Review Exercises (p. 81–83)
- Section 2.1 (p. 85–105)
- Section 2.2 (p. 105–118)
- Section 2.3 (p. 118–134)
- Section 2.4 (p. 134–147)
- Chapter 2 Review Exercises (p. 148–151)
- Section 3.1 (p. 153–166)
- Section 3.2 (p. 166–177)
- Section 3.3 (p. 177–187)
- Section 3.4 (p. 188–206)
- Chapter 3 Review Exercises (p. 206–210)
- Section 4.1 (p. 213–225)
- Section 4.2 (p. 225–238)
- Section 4.3 (p. 238–250)
- Section 4.4 (p. 250–267)
- Section 4.5 (p. 268–281)
- Section 4.6 (p. 281–292)
- Section 4.7 (p. 292–311)
- Chapter 4 Review Exercises (p. 311–317)
- Section 5.1 (p. 319–335)
- Section 5.2 (p. 336–353)
- Section 5.3 (p. 353–365)
- Section 5.4 (p. 365–373)
- Section 5.5 (p. 374–387)
- Section 5.6 (p. 388–406)
- Chapter 5 Review Exercises (p. 406–412)
- Section 6.1 (p. 413–428)
- Section 6.2 (p. 428–445)
- Section 6.3 (p. 445–460)
- Chapter 6 Review Exercises (p. 460–463)
- Section 7.1 (p. 465–473)
- Section 7.2 (p. 473–480)
- Section 7.3 (p. 480–483)
- Section 7.4 (p. 484–491)
- Section 7.5 (p. 492–509)
- Chapter 7 Review Exercises (p. 509–512)
- Section 8.1 (p. 513–527)
- Section 8.2 (p. 527–530)
- Section 8.3 (p. 530–540)
- Section 8.4 (p. 540–552)
- Section 8.5 (p. 552–557)
- Section 8.6 (p. 557–564)
- Section 8.7 (p. 564–581)
- Section 8.8 (p. 581–590)
- Section 8.9 (p. 591–598)
- Section 8.10 (p. 598–605)
- Section 9.1 (p. 607–620)
- Section 9.2 (p. 621–629)
- Section 9.3 (p. 629–635)
- Section 9.4 (p. 636–644)
- Section 9.5 (p. 644–666)
- Appendix B (p. 675–685)
- Appendix C (p. 687–691)
- Appendix D (p. 693–700)

- Index
- Inside Back Cover
- Equivalent Conditions for Linearly Independent and Linearly Dependent Sets
- Kernel Method (Finding a Basis for the Kernel of
*L*) - Range Method (Finding a Basis for the Range of
*L*) - Equivalent Conditions for One-to-One, Onto, and Isomorphism
- Dimension Theorem
- Gram-Schmidt Process

## Details

- No. of pages:
- 806

- Language:
- English

- Copyright:
- © Academic Press 2016

- Published:
- 25th February 2016

- Imprint:
- Academic Press

- Hardcover ISBN:
- 9780128008539

- eBook ISBN:
- 9780128010471

## About the Authors

### Stephen Andrilli

Dr. Stephen Andrilli holds a Ph.D. degree in mathematics from Rutgers University, and is an Associate Professor in the Mathematics and Computer Science Department at La Salle University in Philadelphia, PA, having previously taught at Mount St. Mary’s University in Emmitsburg, MD. He has taught linear algebra to sophomore/junior mathematics, mathematics-education, chemistry, geology, and other science majors for over thirty years. Dr. Andrilli’s other mathematical interests include history of mathematics, college geometry, group theory, and mathematics-education, for which he served as a supervisor of undergraduate and graduate student-teachers for almost two decades. He has pioneered an Honors Course at La Salle based on Douglas Hofstadter’s “Godel, Escher, Bach,” into which he weaves the Alice books by Lewis Carroll. Dr. Andrilli lives in the suburbs of Philadelphia with his wife Ene. He enjoys travel, classical music, classic movies, classic literature, science-fiction, and mysteries. His favorite author is J. R. R. Tolkien.

### Affiliations and Expertise

LaSalle University, Philadelphia, PA, USA

### David Hecker

Dr. David Hecker has a Ph.D. degree in mathematics from Rutgers University, and is a Professor in the Mathematics Department at Saint Joseph’s University in Philadelphia, PA. He has taught linear algebra to sophomore/junior mathematics and science majors for over three decades. Dr. Hecker has previously served two terms as Chair of his department, and his other mathematical interests include real and complex analysis, and linear algebra. He lives on five acres in the farmlands of New Jersey with his wife Lyn, and is very devoted to his four children. Dr. Hecker enjoys photography, camping and hiking, beekeeping, geocaching, science-fiction, humorous jokes and riddles, and rock and country music. His favorite rock group is the Moody Blues.

### Affiliations and Expertise

Saint Joseph's University, Philadelphia, PA, USA

## Reviews

"...they succeed at building an algorithmic feel for the subject, with many summaries of how to do things like diagonalizing a matrix in six easy steps or testing a set of vectors for linear independence in three steps...allows the reader to choose more practice when desired, without it being necessary to complete every exercise." **--MAA Reviews**