LINEAR FUNCTIONS AND MATRIX THEORY

Transcription

1 LINEAR FUNCTIONS AND MATRIX THEORY

2 LINEAR FUNCTIONS AND MATRIX THEORY Bill}acob Department of Mathematics University of California, Santa Barbara Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest

3 Textbooks in Mathematical Sciences Series Editors: Thomas F. Banchoff Brown University John Ewing Indiana University Gaston Gonnet Em Zentrum, Zurich Jerrold Marsden University of California, Berkeley Stan Wagon Macalester College COVER: Paul Kiee, Beware of Red, Private collection, Milan, Italy. Used by permission of Erich Lessing! Art Resource, NY. Grateful acknowledgment is also given for permission to use the following: p. 65: Albrecht Diirer, Albrectus Durerus Nurembergensis Pictor Nuius..., Paris, 1532, "Demonstration of perspective" (woodcut), Spencer Collection, The New York Public Ubrary, Astor, Lenox, and Tilden Foundations; p. 70: Albrecht Durer, The Nativity (1504), The Metropolitan Museum of Art, Fletcher Fund, Ubrary of Congress Cataloging-In-Publication Data Jacob, Bill. linear functions and matrix theory / Bill Jacob. p. cm. - (fextbooks In mathematical sciences) Includes bibliographical references and Index. ISBN-13: e-isbn-i3: : / Algebras, linear. 2. Matrices. I. Title. II. Series. QA184.) '.5-dc Printed on acid-free paper Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or In part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts In connection with reviews or scholarly analysis. Use In connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissintilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., In this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Steven Pisano; manufacturing supervised by Jacqui Ashri. Photocomposed by Integre Technical Publishing Co., Inc. from the author's latex files

4 PREFACE Courses that study vectors and elementary matrix theory and introduce linear transformations have proliferated greatly in recent years. Most of these courses are taught at the undergraduate level as part of, or adjacent to, the second-year calculus sequence. Although many students will ultimately find the material in these courses more valuable than calculus, they often experience a class that consists mostly of learning to implement a series of computational algorithms. The objective of this text is to bring a different vision to this course, including many of the key elements called for in current mathematics-teaching reform efforts. Three of the main components of this current effort are the following: 1. Mathematical ideas should be introduced in meaningful contexts, with formal definitions and procedures developed after a clear understanding of practical situations has been achieved. 2. Every topic should be treated from different perspectives, including the numerical, geometric, and symbolic viewpoints. 3. The important ideas need to be visited repeatedly throughout the term, with students' understan9ing deepening each time. This text was written with these three objectives in mind. The first two chapters deal with situations requiring linear functions (at times, locally linear functions) or linear ideas in geometry for their understanding. These situations provide the context in which the formal mathematics is developed, and they are returned to with increasing sophistication throughout the text. In addition, expectations of student work have changed. Computer technology has reduced the need for students to devote large blocks of time learning to implement computational algorithms. Instead, we demand a deeper conceptual understanding. Students need to learn how to communicate mathematics effectively, both orally and in writing. Students also need how to learn to use technology, applying it when appropriate and giving meaningful answers with it. Further, students need to collaborate on mathematical problems, and thus this collaboration often involves mathematical investigations where the final outcome depends on the assumptions they make. This text'is designed to provide students with the opportunity to develop their skills in each of these areas. There are ample computational v

5 vi PREFACE exercises so that students can develop familiarity with all the basic algorithms of the subject. However, many problems are not preceded by a worked-out example of a similar-looking problem, so students must spend some time grappling with the concepts rather than mimicking procedures. Each section concludes with problems or projects designed for student collaborative work. There is quite a bit of variation in the nature of these group projects, and most of them require more discussion and struggle than the regular problems. A number of them are open-ended, without single answers, and therefore part of the project is finding how to formulate the question so that the mathematics can be applied. Throughout the text, as well as in the problems, there are many occasions where technology is needed for calculation. Most college students have access to graphing calculators, and the majority of these calculators are capable of performing all of the basic matrix calculations needed for the use of this text. Students should be encouraged to use technology where appropriate, and if a computer laboratory is available it will be useful, too. Some problems explicitly require calculators or use of a computer; others clearly do not; and on some occasions the student needs to take the initiative to make an intelligent use of technology. During the in-class testing of the material (as part of a second-year calculus sequence), instructors found that the use of graphing calculators gave students more time to focus on conceptual aspects of the material. Instructors used to assigning a large volume of algorithmic exercises found they had to reduce the number of problems assigned so that students had more opportunity to explore the ideas in the problems. A brief outline of how the text is organized follows. The first six chapters constitute a core course covering the material usually taught as part of a second-year sequence. Depending on how the class is paced, these chapters require seven to ten weeks to cover. The remaining three chapters deal with more advanced topics. Although they are designed to be taught in sequence, their order can be varied provided the instructor is willing to explain a few results from the omitted sections. Taken together, all nine chapters have more than enough material for a full-semester introductory course. Sections 1.4, 2.4, 3.5, and 4.5 can be omitted if time is tight, although it would be preferable to avoid this, since their purpose is to give geometric meaning to material that students too often view purely symbolically. Chapter 6 could be covered immediately after Chap. 2 should the instructor prefer. No knowledge of calculus is assumed in the body of the text. However, some group projects are designed for use by students familiar with calculus. Answers to the odd-numbered problems are given at the end of the text. Chapter 1 introduces the concept of a linear function. The main purpose of the chapter is to illustrate the numerical and geometric meaning of linearity and local linearity for functions. In Sec. 1.2, real data are analyzed

6 PREFACE vii so that students see why this subject was developed. Matrices arise, initially, for convenience of notation. Linearly constrained maxima and minima problems are introduced because the study of level sets in this context provides one of the best ways to illustrate the geometric meaning of linearity for functions of several variables. Chapter 2 studies the linear geometry of lines and planes in two- and three-dimensional space by considering problems that require their use. A main goal here is to provide a familiar geometric setting for introducing the use of vectors and matrix notation. The last section, covering linear perspective, illustrates how these topics impact our daily lives by showing how three-dimensional objects are represented on the plane. Chapter 3 develops the basic principles of Gaussian and Gauss-jordan elimination and their use in solving systems of linear equations. Matrix rank is studied in the context of understanding the structure of solutions to systems of equations. Some basic problems in circuit theory motivate the study of systems of equations. The simplex algorithm is introduced in the last section, illustrating how the ideas behind Gaussian elimination can be used to solve the constrained optimization problems introduced geometrically in Chap. l. Chapter 4 treats basic matrix algebra and its connections with systems of linear equations. The use of matrices in analyzing iterative processes, such as Markov chains or Fibonacci numbers, provides the setting for the development of matrix properties. The determinant is developed using the the Laplace expansion, and applications including the adjoint inversion formula and Cramer's rule are given. The chapter concludes with discussion of the LU-decomposition and its relationship to Gaussian elimination, determinants, and tridiagonal matrices. Chapter 5 develops the basic linear algebra concepts of linear combinations, linear independence, subspaces, span, and dimension. Problems involving network flow and stoichiometry are considered and provide background for why these basic linear algebra concepts are so important. All of these topics are treated in the setting of R n only, although the results are formulated in such a way that the proofs apply to general vector spaces. Chapter 6 returns to more vector geometry in two- and three-dimensional space. The emphasis is on applying the dot and cross product in answering geometric questions. The geometry of how carbon atoms fit together in cyclohexane ring systems is studied to help develop three-dimensional visual thinking. As mentioned, this material could be covered immediately after Chap. 2 if the instructor chooses. The author, however, prefers to have his students study this chapter after Chap. 5 in order to remind them of the importance of geometric thinking. Chapter 7 studies eigenvalues and eigenvectors and their role in the problem of diagonalizing matrices. Motivation for considering eigenvec-

7 viii PREFACE tors is provided by the return to the study of iterative processes initiated in Chap. 4. The cases of symmetric and probability matrices are studied in detail. This material is developed from a matrix perspective, prior to the treatment of linear operators (Chap. 8), for instructors who like to get to this topic as early as possible. In fact, this material provides nice motivation for Chap. 8. Chapter 8 develops the theory of linear transformations and matrix representations of linear operators on Rn. A main objective is to show how the point of view of linear transformations unifies many of the matrixoriented subjects treated earlier. The chapter returns to the examples of electrical networks first studied in Chap. 3, where the cascading of networks provides a basis for understanding the composition of linear transformations. The basic geometric transformations of rotations and reflections are also studied. Chapter 9 returns to the geometry of Euclidean space. The Gram Schmidt process and orthogonal projections can be found here. Leastsquares problems are also studied from the geometric point of view. Some of the data given in Chap. 1 are fit using linear regressions, bringing the course to a close by showing how the concepts developed in the class deepen our understanding of some of the original problems considered. I would like to thank my numerous students and colleagues for their valuable input during the development of this text, provided both anonymously and in person. I am especially grateful to Juan Estrada, Gustavo Ponce, and Gerald Zaplawa for their detailed comments. I would also like to thank Jerry Lyons of Springer-Verlag for his encouragement and support. Finally, I wish to thank my family for their love throughout this project. Bill Jacob Santa Barbara, California

8 CONTENTS Preface v 1 Linear Functions 1.1 Linear Functions Local Linearity Matrices More Linearity 30 2 Linear Geometry 2.1 Linear Geometry in the Plane Vectors and Lines in the Plane Linear Geometry in Space An Introduction to Linear Perspective 64 3 Systems of Linear Equations 3.1 Systems of Linear Equations Gaussian Elimination Gauss-Jordan Elimination Matrix Rank and Systems of Linear Equations The Simplex Algorithm Basic Matrix Algebra 4.1 The Matrix Product: A Closer Look....., Fibonacci Numbers and Difference Equations The Determinant Properties and Applications of the Determinant The LU-Decomposition Key Concepts of Linear Algebra in R" 5.1 Linear Combinations and Subspaces Linear Independence Basis and Dimension Ix

9 x CONTENTS 6 More Vector Geometry 6.1 The Dot Product Angles and Projections The Cross Product Eigenvalues and Eigenvectors of Matrices 7.1 Eigenvalues and Eigenvectors Eigenspaces and Diagonalizability Symmetric Matrices and Probability Matrices Matrices as linear Transformations 8.1 Linear Transformations Using Linear Transformations Change of Basis Orthogonality and Least-Squares Problems 9.1 Orthogonality and the Gram-Schmidt Process Orthogonal Projections Least-Squares Approximations Answers to Odd-Numbered Problems 303 Index 327

10 LINEAR FUNCTIONS AND MATRIX TIlEORY