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  • ISBN 978-0-07-353235-6 MHID 0-07-353235-5

    Linear Algebra

    Introduction to Linear Algebra with Applications by Jim DeFranza and Daniel Gagliardi provides the proper balance between computation, problem solving, and abstraction that will equip students with the necessary skills and problem solving strategies to allow for a greater understanding and appreciation of linear algebra and its numerous applications.

    Introduction to Linear Algebra with Applications provides students with the necessary tools for success:

    Abstract theory is essential to understanding how linear algebra is applied.

    Each concept is fully developed presenting natural connections between topics giving students a working knowledge of the theory and techniques for each module covered.

    Applications have been carefully chosen to highlight the utility of linear algebra in order to see the relevancy of the subject matter in other areas of science as well as in mathematics.

    Ranging from routine to more challenging, each exercise set extends the concepts or techniques by asking the student to construct complete arguments. End of chapter True/False questions help students connect concepts and facts presented in the chapter.

    Examples are designed to develop intuition and prepare students to think more conceptually about new topics as they are introduced.

    Students are introduced to the study of linear algebra in a sequential and thorough manner through an engaging writing style gaining a clear understanding of the theory essential for applying linear algebra to mathematics or other � elds of science.

    Summaries conclude each section with important facts and techniques providing students with easy access to the material needed to master the exercise sets.

    Linear AlgebraLinear Algebra DeFranza

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  • Revised Pages



    Jim DeFranza St. Lawrence University

    Dan Gagliardi SUNY Canton

  • First Pages


    Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright  2009 by The McGraw-Hill Companies, Inc. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning.

    Some ancillaries, including electronic and print components, may not be available to customers outside the United States.

    This book is printed on acid-free paper.

    1 2 3 4 5 6 7 8 9 0 DOC/DOC 0 9 8

    ISBN 978–0–07–353235–6 MHID 0–07–353235–5

    Editorial Director: Stewart K. Mattson Senior Sponsoring Editor: Elizabeth Covello Director of Development: Kristine Tibbetts Developmental Editor: Michelle Driscoll Director of Marketing: Ryan Blankenship Marketing Coordinator: Sabina Navsariwala Project Manager: Joyce Watters Senior Production Supervisor: Sherry L. Kane Senior Media Project Manager: Tammy Juran Designer: Laurie B. Janssen Cover Designer: Ron Bissell (USE) Cover Image: Royalty-Free/CORBIS Senior Photo Research Coordinator: John C. Leland Supplement Producer: Melissa M. Leick Compositor: Laserwords Private Limited Typeface: 10.25/12 Times Printer: R. R. Donnelly Crawfordsville, IN

    Library of Congress Cataloging-in-Publication Data

    DeFranza, James, 1950– Introduction to linear algebra / James DeFranza, Daniel Gagliardi. —1st ed.

    p. cm. Includes index. ISBN 978–0–07–353235–6—ISBN 0–07–353235–5 (hard copy : alk. paper)

    1. Algebras, Linear—Textbooks. 2. Algebras, Linear—Problems, exercises, etc. I. Gagliardi, Daniel. II. Title.

    QA184.2.D44 2009 515′ .5—dc22


  • Revised Pages

    To Regan, Sara, and David


    To Robin, Zachary, Michael, and Eric


  • Revised Pages

  • About the Authors

    Jim DeFranza was born in 1950 in Yonkers New York and grew up in Dobbs Ferry New York on the Hudson River. Jim DeFranza is Professor of Mathematics at St. Lawrence University in Canton New York where he has taught undergraduate mathematics for 25 years. St. Lawrence University is a small Liberal Arts College in upstate New York that prides itself in the close interaction that exists between students and faculty. It is this many years of working closely with students that has shaped this text in Linear Algebra and the other texts he has written. He received his Ph.D. in Pure Mathematics from Kent State University in 1979. Dr. DeFranza has coauthored PRECALCULUS, Fourth Edition and two other texts in single variable and multivariable calculus. Dr. DeFranza has also published a dozen research articles in the areas of Sequence Spaces and Classical Summability Theory. Jim is married and has two children David and Sara. Jim and his wife Regan live outside of Canton New York in a 150 year old farm house.

    Daniel Gagliardi is an Assistant Professor of Mathematics at SUNY Canton, in Canton New York. Dr. Gagliardi began his career as a software engineer at IBM in East Fishkill New York writing programs to support semiconductor development and manufacturing. He received his Ph.D. in Pure Mathematics from North Carolina State University in 2003 under the supervision of Aloysius Helminck. Dr. Gagliardi’s principle area of research is in Symmetric Spaces. In particular, his current work is concerned with developing algorithmic formulations to describe the fine structure (characters and Weyl groups) of local symmetric spaces. Dr. Gagliardi also does research in Graph Theory. His focus there is on the graphical realization of certain types of sequences. In addition to his work as a mathematician, Dr. Gagliardi is an accomplished double bassist and has recently recorded a CD of jazz standards with Author/Pianist Bill Vitek. Dr. Gagliardi lives in northern New York in the picturesque Saint Lawrence River Valley with his wife Robin, and children Zachary, Michael, and Eric.


    Revised Confirming Pages

  • Revised Pages


    Preface ix

    CHAPTER 1 Systems of Linear Equations andMatrices 1 1.1 Systems of Linear Equations 2

    Exercise Set 1.1 12

    1.2 Matrices and Elementary Row Operations 14 Exercise Set 1.2 23

    1.3 Matrix Algebra 26 Exercise Set 1.3 37

    1.4 The Inverse of a Square Matrix 39 Exercise Set 1.4 45

    1.5 Matrix Equations 48 Exercise Set 1.5 51

    1.6 Determinants 54 Exercise Set 1.6 65

    1.7 Elementary Matrices and LU Factorization 68 Exercise Set 1.7 77

    1.8 Applications of Systems of Linear Equations 79 Exercise Set 1.8 84

    Review Exercises 89

    Chapter Test 90

    CHAPTER 2 Linear Combinations and Linear Independence 93 2.1 Vectors in �n 94

    Exercise Set 2.1 99

    2.2 Linear Combinations 101 Exercise Set 2.2 108

    2.3 Linear Independence 111 Exercise Set 2.3 120

    Review Exercises 123

    Chapter Test 125


  • Revised Pages

    Contents vii

    CHAPTER 3 Vector Spaces 127 3.1 Definition of a Vector Space 129

    Exercise Set 3.1 137

    3.2 Subspaces 140 Exercise Set 3.2 154

    3.3 Basis and Dimension 156 Exercise Set 3.3 171

    3.4 Coordinates and Change of Basis 173 Exercise Set 3.4 182

    3.5 Application: Differential Equations 185 Exercise Set 3.5 193

    Review Exercises 194

    Chapter Test 195

    CHAPTER 4 Linear Transformations 199 4.1 Linear Transformations 200

    Exercise Set 4.1 211

    4.2 The Null Space and Range 214 Exercise Set 4.2 223

    4.3 Isomorphisms 226 Exercise Set 4.3 233

    4.4 Matrix Representation of a Linear Transformation 235 Exercise Set 4.4 245

    4.5 Similarity 249 Exercise Set 4.5 253

    4.6 Application: Computer Graphics 255 Exercise Set 4.6 268

    Review Exercises 270

    Chapter Test 272

    CHAPTER 5 Eigenvalues and Eigenvectors 275 5.1 Eigenvalues and Eigenvectors 276

    Exercise Set 5.1 285

    5.2 Diagonalization 287 Exercise Set 5.2 298

    5.3 Application: Systems of Linear Differential Equations 300 Exercise Set 5.3 309

  • Revised Pages

    viii Contents

    5.4 Application: Markov Chains 310 Exercise Set 5.4 315

    Review Exercises 316

    Chapter Test 318

    CHAPTER 6 Inner Product Spaces 321 6.1 The Dot Product on �n 323

    Exercise Set 6.1 331

    6.2 Inner Product Spaces 333 Exercise Set 6.2 341

    6.3 Orthonormal Bases 342 Exercise Set 6.3 352

    6.4 Orthogonal Complements 355 Exercise Set 6.4 364

    6.5 Application: Least Squares Approximation 366 Exercise Set 6.5 375

    6.6 Diagonalization of Symmetric Matrices 377 Exercise Set 6.6 383

    6.7 Application: Quadratic Forms 385 Exercise Set 6.7 392

    6.8 Application: Singular Value Decomposition 392 Exercise Set 6.8 403

    Review Exercises 404

    Chapter Test 406

    Appendix 409 Answers to Odd-Numbered Exercises 440 Index 479

  • Revised Pages


    Introduction to Linear Algebra with Applications is an introductory text targeted to second-y


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