org About The Author Howard Anton obtained his B. from Lehigh University, his M. from the University of Illinois, and his Ph. from the Polytechnic University of Brooklyn, all in mathematics.
In the early 1960s he worked for Burroughs Corporation and Avco Corporation at Cape Canaveral, Florida, where he was involved with the manned space program. In 1968 he joined the Mathematics Department at Drexel University, where he taught full time until 1983. Since then he has devoted the majority of his time to textbook writing and activities for mathematical associations. Anton was president of the EPADEL Section of the Mathematical Association of America (MAA), served on the Board of Governors of that organization, and guided the creation of the Student Chapters of the MAA.
In addition to various pedagogical articles, he has published numerous research papers in functional analysis, approximation theory, and topology. He is best known for his textbooks in mathematics, which are among the most widely used in the world. There are currently more than 150 versions of his books, including translations into Spanish, Arabic, Portuguese, Italian, Indonesian, French, Japanese, Chinese, Hebrew, and German. For relaxation, Dr.
Anton enjoys travel and photography. Copyright © 2010 John Wiley & Sons, Inc. All rights reserved. Preface This edition of Elementary Linear Algebra gives an introductory treatment of linear algebra that is suitable for a first undergraduate course.
Its aim is to present the fundamentals of linear algebra in the clearest possible way—sound pedagogy is the main consideration. Although calculus is not a prerequisite, there is some optional material that is clearly marked for students with a calculus background. If desired, that material can be omitted without loss of continuity. Technology is not required to use this text, but for instructors who would like to use MATLAB, Mathematica, Maple, or calculators with linear algebra capabilities, we have posted some supporting material that can be accessed at either of the following Web sites: www.com/college/anton Summary of Changes in this Edition This edition is a major revision of its predecessor.
In addition to including some new material, some of the old material has been streamlined to ensure that the major topics can all be covered in a standard course. These are the most significant changes: • Vectors in 2-space, 3-space, and n-space Chapters 3 and 4 of the previous edition have been combined into a single chapter. This has enabled us to eliminate some duplicate exposition and to juxtapose concepts in n-space with those in 2-space and 3-space, thereby conveying more clearly how n-space ideas generalize those already familiar to the student. • New Pedagogical Elements Each section now ends with a Concept Review and a Skills mastery that provide the student a convenient reference to the main ideas in that section.
• New Exercises Many new exercises have been added, including a set of True/False exercises at the end of most sections. • Earlier Coverage of Eigenvalues and Eigenvectors The chapter on eigenvalues and eigenvectors, which was Chapter 7 in the previous edition, is Chapter 5 in this edition. • Complex Vector Spaces The chapter entitled Complex Vector Spaces in the previous edition has been completely revised. The most important ideas are now covered in Section 5.5 in the context of matrix diagonalization.
A brief review of complex numbers is included in the Appendix. • Quadratic Forms This material has been extensively rewritten to focus more precisely on the most important ideas. • New Chapter on Numerical Methods In the previous edition an assortment of topics appeared in the last chapter. That chapter has been replaced by a new chapter that focuses exclusively on numerical methods of linear algebra.
We achieved this by moving those topics not concerned with numerical methods elsewhere in the text. • Singular-Value Decomposition In recognition of its growing importance, a new section on Singular-Value Decomposition has been added to the chapter on numerical methods.org • Internet Search and the Power Method A new section on the Power Method and its application to Internet search engines has been added to the chapter on numerical methods. • Applications There is an expanded version of this text by Howard Anton and Chris Rorres entitled Elementary Linear Algebra: Applications Version, 10th (ISBN 9780470432051), whose purpose is to supplement this version with an extensive body of applications. However, to accommodate instructors who asked us to include some applications in this version of the text, we have done so.
These are generally less detailed than those appearing in the Anton/Rorres text and can be omitted without loss of continuity. Hallmark Features • Relationships Among Concepts One of our main pedagogical goals is to convey to the student that linear algebra is a cohesive subject and not simply a collection of isolated definitions and techniques. One way in which we do this is by using a crescendo of Equivalent Statements theorems that continually revisit relationships among systems of equations, matrices, determinants, vectors, linear transformations, and eigenvalues. To get a general sense of how we use this technique see Theorems 1.4 and then Theorem 5.
• Smooth Transition to Abstraction Because the transition from Rn to general vector spaces is difficult for many students, considerable effort is devoted to explaining the purpose of abstraction and helping the student to “visualize” abstract ideas by drawing analogies to familiar geometric ideas. • Mathematical Precision When reasonable, we try to be mathematically precise. In keeping with the level of student audience, proofs are presented in a patient style that is tailored for beginners. There is a brief section in the Appendix on how to read proof statements, and there are various exercises in which students are guided through the steps of a proof and asked for justification.
• Suitability for a Diverse Audience This text is designed to serve the needs of students in engineering, computer science, biology, physics, business, and economics as well as those majoring in mathematics. • Historical Notes To give the students a sense of mathematical history and to convey that real people created the mathematical theorems and equations they are studying, we have included numerous Historical Notes that put the topic being studied in historical perspective. About the Exercises • Graded Exercise Sets Each exercise set begins with routine drill problems and progresses to problems with more substance. • True/False Exercises Most exercise sets end with a set of True/False exercises that are designed to check conceptual understanding and logical reasoning.
To avoid pure guessing, the students are required to justify their responses in some way. • Supplementary Exercise Sets Most chapters end with a set of supplementary exercises that tend to be more challenging and force the student to draw on ideas from the entire chapter rather than a specific section. Supplementary Materials for Students • Student Solutions Manual This supplement provides detailed solutions to most theoretical exercises and to at least one nonroutine exercise of every type (ISBN 9780470458228). • Technology Exercises and Data Files The technology exercises that appeared in the previous edition have been moved to the Web site that accompanies this text.
Those exercises are designed to be solved using MATLAB, Mathematica, or Maple and are accompanied by data files in all three formats. The exercises and data can be downloaded from either of the following Web sites.com/college/anton Supplementary Materials for Instructors • Instructor's Solutions Manual This supplement provides worked-out solutions to most exercises in the text (ISBN 9780470458235). • WileyPLUS™ This is Wiley's proprietary online teaching and learning environment that integrates a digital version of this textbook with instructor and student resources to fit a variety of teaching and learning styles. WileyPLUS will help your students master concepts in a rich and structured environment that is available to them 24/7.
It will also help you to personalize and manage your course more effectively with student assessments, assignments, grade tracking, and other useful tools. • Your students will receive timely access to resources that address their individual needs and will receive immediate feedback and remediation resources when needed. • There are also self-assessment tools that are linked to the relevant portions of the text that will enable your students to take control of their own learning and practice. • WileyPLUS will help you to identify those students who are falling behind and to intervene in a timely manner without waiting for scheduled office hours.
More information about WileyPLUS can be obtained from your Wiley representative. A Guide for the Instructor Although linear algebra courses vary widely in content and philosophy, most courses fall into two categories —those with about 35–40 lectures and those with about 25–30 lectures. Accordingly, we have created long and short templates as possible starting points for constructing a course outline. Of course, these are just guides, and you will certainly want to customize them to fit your local interests and requirements.
Neither of these sample templates includes applications. Those can be added, if desired, as time permits. Long Template Short Template Chapter 1: Systems of Linear Equations and Matrices 7 lectures 6 lectures Chapter 2: Determinants 3 lectures 2 lectures Long Template Short Template Chapter 3: Euclidean Vector Spaces 4 lectures 3 lectures Chapter 4: General Vector Spaces 10 lectures 10 lectures Chapter 5: Eigenvalues and Eigenvectors 3 lectures 3 lectures Chapter 6: Inner Product Spaces 3 lectures 1 lecture Chapter 7: Diagonalization and Quadratic Forms 4 lectures 3 lectures Chapter 8: Linear Transformations 3 lectures 2 lectures Total: 37 lectures 30 lectures Acknowledgements I would like to express my appreciation to the following people whose helpful guidance has greatly improved the text. Reviewers and Contributors Don Allen, Texas A&M University John Alongi, Northwestern University John Beachy, Northern Illinois University Przemslaw Bogacki, Old Dominion University Robert Buchanan, Millersville University of Pennsylvania Ralph Byers, University of Kansas Evangelos A.
Coutsias, University of New Mexico Joshua Du, Kennesaw State University Fatemeh Emdad, Michigan Technological University Vincent Ervin, Clemson University Anda Gadidov, Kennesaw State University Guillermo Goldsztein, Georgia Institute of Technology Tracy Hamilton, California State University, Sacramento Amanda Hattway, Wentworth Institute of Technology Heather Hulett, University of Wisconsin—La Crosse David Hyeon, Northern Illinois University Matt Insall, Missouri University of Science and Technology Mic Jackson, Earlham College Anton Kaul, California Polytechnic Institute, San Luis Obispo www.org Harihar Khanal, Embry-Riddle University Hendrik Kuiper, Arizona State University Kouok Law, Georgia Perimeter College James McKinney, California State University, Pomona Eric Schmutz, Drexel University Qin Sheng, Baylor University Adam Sikora, State University of New York at Buffalo Allan Silberger, Cleveland State University Dana Williams, Dartmouth College Mathematical Advisors Special thanks are due to a number of talented teachers and mathematicians who provided pedagogical guidance, provided help with answers and exercises, or provided detailed checking or proofreading: John Alongi, Northwestern University Scott Annin, California State University, Fullerton Anton Kaul, California Polytechnic State University Sarah Streett Cindy Trimble, C Trimble and Associates Brad Davis, C Trimble and Associates The Wiley Support Team David Dietz, Senior Acquisitions Editor Jeff Benson, Assistant Editor Pamela Lashbrook, Senior Editorial Assistant Janet Foxman, Production Editor Maddy Lesure, Senior Designer Laurie Rosatone, Vice President and Publisher Sarah Davis, Senior Marketing Manager Diana Smith, Marketing Assistant Melissa Edwards, Media Editor Lisa Sabatini, Media Project Manager Sheena Goldstein, Photo Editor Carol Sawyer, Production Manager Lilian Brady, Copyeditor Special Contributions The talents and dedication of many individuals are required to produce a book such as this, and I am fortunate to have benefited from the expertise of the following people: David Dietz — my editor, for his attention to detail, his sound judgment, and his unwavering faith in me. Jeff Benson — my assistant editor, who did an unbelievable job in organizing and coordinating the many threads required to make this edition a reality. Carol Sawyer — of The Perfect Proof, who coordinated the myriad of details in the production process. It will be a pleasure to finally delete from my computer the hundreds of emails we exchanged in the course of working together on this book.