com Elementary Linear Algebra Fourth Edition Stephen Andrilli Department of Mathematics and Computer Science La Salle University Philadelphia, PA David Hecker Department of Mathematics Saint Joseph’s University Philadelphia, PA AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier www.com Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK Copyright © 2010 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.
You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data Application submitted. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN: 978-0-12-374751-8 For information on all Academic Press publications visit our Web site at www.com Printed in Canada 09 10 11 9 8 7 6 5 4 3 2 1 www.com To our wives, Ene and Lyn, for all their help and encouragement www.com This page intentionally left blank www.com Contents Preface for the Instructor. ix Preface for the Student.
xix Symbol Table. xxiii Computational and Numerical Methods, Applications. xxvii CHAPTER 1 Vectors and Matrices 1 1.1 Fundamental Operations with Vectors .2 The Dot Product.3 An Introduction to Proof Techniques .4 Fundamental Operations with Matrices. 59 CHAPTER 2 Systems of Linear Equations 79 2.1 Solving Linear Systems Using Gaussian Elimination .2 Gauss-Jordan Row Reduction and Reduced Row Echelon Form .3 Equivalent Systems, Rank, and Row Space.4 Inverses of Matrices.
125 CHAPTER 3 Determinants and Eigenvalues 143 3.1 Introduction to Determinants .2 Determinants and Row Reduction .3 Further Properties of the Determinant .4 Eigenvalues and Diagonalization. 178 CHAPTER 4 Finite Dimensional Vector Spaces 203 4.1 Introduction to Vector Spaces .5 Basis and Dimension .6 Constructing Special Bases. 281 CHAPTER 5 Linear Transformations 305 5.1 Introduction to Linear Transformations.2 The Matrix of a Linear Transformation .com vi Contents 5.3 The Dimension Theorem .4 One-to-One and Onto Linear Transformations .6 Diagonalization of Linear Operators .1 Orthogonal Bases and the Gram-Schmidt Process. 428 CHAPTER 7 Complex Vector Spaces and General Inner Products 445 7.1 Complex n-Vectors and Matrices .2 Complex Eigenvalues and Complex Eigenvectors .3 Complex Vector Spaces .4 Orthogonality in Cn .5 Inner Product Spaces.
472 CHAPTER 8 Additional Applications 491 8.3 Least-Squares Polynomials .5 Hill Substitution: An Introduction to Coding Theory .7 Rotation of Axes for Conic Sections .10 Least-Squares Solutions for Inconsistent Systems. 578 CHAPTER 9 Numerical Methods 587 9.1 Numerical Methods for Solving Systems .3 The Power Method for Finding Eigenvalues .5 Singular Value Decomposition. 623 Appendix A Miscellaneous Proofs 645 Proof of Theorem 1. 645 Proof of Theorem 2.
646 Proof of Theorem 2.com Contents vii Proof of Theorem 3. 648 Proof of Theorem 5. 649 Proof of Theorem 6. 650 Appendix B Functions 653 Functions: Domain, Codomain, and Range.
653 One-to-One and Onto Functions. 654 Composition and Inverses of Functions. 655 Appendix C Complex Numbers 661 Appendix D Answers to Selected Exercises 665 Index 725 www.com This page intentionally left blank www.com Preface for the Instructor This textbook is intended for a sophomore- or junior-level introductory course in linear algebra. We assume the students have had at least one course in calculus.
PHILOSOPHY AND FEATURES OF THE TEXT Clarity of Presentation: We have striven for clarity and used straightforward lan- guage throughout the book, occasionally sacrificing brevity for clear and convincing explanation. We hope you will encourage students to read the text deeply and thoroughly. Helpful Transition from Computation to Theory: In writing this text, our main intention was to address the fact that students invariably ran into trouble as the largely com- putational first half of most linear algebra courses gave way to a more theoretical second half. In particular,many students encountered difficulties when abstract vector space topics were introduced.
Accordingly, we have taken great care to help students master these important concepts. We consider the material in Sections 4.6 (vector spaces and subspaces, span, linear independence, basis and dimension, coordinatization, linear transformations, kernel and range, one-to-one and onto linear transformations, isomorphism, diagonalization of linear operators) to be the “heart” of this linear algebra text. Emphasis on the Reading and Writing of Proofs: One reason that students have trouble with the more abstract material in linear algebra is that most textbooks contain few, if any, guidelines about reading and writing simple mathematical proofs. This book is intended to remedy that situation.
Consequently, we have students working on proofs as quickly as possible. After a discussion of the basic properties of vectors, there is a special section (Section 1.3) on general proof techniques, with concrete exam- ples using the material on vectors from Sections 1. The early placement of Section 1.3 helps to build the students’confidence and gives them a strong foundation in the reading and writing of proofs. We have written the proofs of theorems in the text in a careful manner to give students models for writing their own proofs.
We avoided “clever” or “sneaky” proofs, in which the last line suddenly produces “a rabbit out of a hat,” because such proofs invariably frustrate students. They are given no insight into the strategy of the proof or how the deductive process was used. In fact, such proofs tend to reinforce the students’ mistaken belief that they will never become competent in the art of writing proofs. In this text,proofs longer than one paragraph are often written in a“top-down” manner, a concept borrowed from structured programming.
A complex theorem is broken down into a secondary series of results, which together are sufficient to prove the original theorem. In this way,the student has a clear outline of the logical argument and can more easily reproduce the proof if called on to do so.com x Preface for the Instructor We have left the proofs of some elementary theorems to the student. However, for every nontrivial theorem in Chapters 1 through 6,we have either included a proof,or given detailed hints which should be sufficient to enable students to provide a proof on their own. Most of the proofs of theorems that are left as exercises can be found in the Student Solutions Manual.The exercises corresponding to these proofs are marked with the symbol .
Computational and Numerical Methods, Applications: A summary of the most important computational and numerical methods covered in this text is found in the chart located in the frontpages. This chart also contains the most important applications of linear algebra that are found in this text. Linear algebra is a branch of mathematics having a multitude of practical applications, and we have included many standard ones so that instructors can choose their favorites. Chapter 8 is devoted entirely to applications of linear algebra, but there are also several shorter applications in Chapters 1 to 6.
Instructors may choose to have their students explore these applications in computer labs,or to assign some of these applications as extra credit reading assignments outside of class. Revisiting Topics: We frequently introduce difficult concepts with concrete examples and then revisit them frequently in increasingly abstract forms as students progress throughout the text. Here are several examples: ■ Students are first introduced to the concept of linear combinations beginning in Section 1.1, long before linear combinations are defined for real vector spaces in Chapter 4. ■ The row space of a matrix is first encountered in Section 2.3, thereby preparing students for the more general concepts of subspace and span in Sections 4.
■ Students traditionally find eigenvalues and eigenvectors to be a difficult topic,so these are introduced early in the text (Section 3.4) in the context of matrices. Further properties of eigenvectors are included throughout Chapters 4 and 5 as underlying vector space concepts are covered. Then a more thorough, detailed treatment of eigenvalues is given in Section 5.6 in the context of linear transfor- mations. The more advanced topics of orthogonal and unitary diagonalization are covered in Chapters 6 and 7.
■ The technique behind the first two methods in Section 4.6 for computing bases are introduced earlier in Sections 4.4 in the Simplified Span Method and the Independence Test Method, respectively. In this way, students will become comfortable with these methods in the context of span and linear independence before employing them to find appropriate bases for vector spaces. ■ Students are first introduced to least-squares polynomials in Section 8.3 in a concrete fashion,and then (assuming a knowledge of orthogonal complements), the theory behind least-squares solutions for inconsistent systems is explored later on in Section 8.com Preface for the Instructor xi Numerous Examples and Exercises: There are 321 numbered examples in the text, and many other unnumbered examples as well, at least one for each new concept or application, to ensure that students fully understand new material before proceeding onward. Almost every theorem has a corresponding example to illustrate its meaning and/or usefulness.
The text also contains an unusually large number of exercises. There are more than 980 numbered exercises, and many of these have multiple parts, for a total of more than 2660 questions. Some are purely computational. Many others ask the students to write short proofs.
The exercises within each section are generally ordered by increasing difficulty, beginning with basic computational problems and moving on to more theoretical problems and proofs. Answers are provided at the end of the book for approximately half the computational exercises; these problems are marked with a star (★). Full solutions to the ★ exercises appear in the Student Solutions Manual. True/False Exercises: Included among the exercises are 500 True/False questions, which appear at the end of each section in Chapters 1 through 9, as well as in the Review Exercises at the end of Chapters 1 through 7, and in Appendices B and C.
These True/False questions help students test their understanding of the fundamental concepts presented in each section. In particular, these exercises highlight the impor- tance of crucial words in definitions or theorems. Pondering True/False questions also helps the students learn the logical differences between “true,” “occasionally true,” and “never true.” Understanding such distinctions is a crucial step toward the type of reasoning they are expected to possess as mathematicians. Summary Tables: There are helpful summaries of important material at various points in the text: ■ Table 2.3): The three types of row operations and their inverses ■ Table 3.2): Equivalent conditions for a matrix to be singular (and similarly for nonsingular) ■ Chart following Chapter 3: Techniques for solving a system of linear equations, and for finding the inverse,determinant,eigenvalues and eigenvectors of a matrix ■ Table 4.4): Equivalent conditions for a subset to be linearly independent (and similarly for linearly dependent) ■ Table 4.6): Contrasts between the Simplified Span Method and the Independence Test Method ■ Table 5.2): Matrices for several geometric linear operators in R3 ■ Table 5.5): Equivalent conditions for a linear transformation to be an isomorphism (and similarly for one-to-one, onto) Symbol Table: Following the Prefaces, for convenience, there is a comprehensive Sym- bol Table listing all of the major symbols related to linear algebra that are employed in this text together with their meanings.com xii Preface for the Instructor Instructor’s Manual: An Instructor’s Manual is available for this text that contains the answers to all computational exercises, and complete solutions to the theoretical and proof exercises.