# [bernard kolman, david_hill]_elementary_linear_alg(book_fi.org)

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LINEAR ALGEBRA

BERNARD KOLMAN DAVID R. HilL

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Edito:1al Director, Computer Science. EIIginecring, and Advanced Mathematics: Marcil, J, Harlon Scnior Editor: Holly Stark EdilO' ial Assistam: Jennifer LonJcheili Scnior Managing EditorlProduction EdilOr: Scoll Diswmo Art Director: Ju,m LOpe! Cove, De~igncr: Mic/rlle/ Fmhm".! Art EdilOr: Thoma.! Bmflmi Manufacturing Buycr: Lisa McDowell Marketing Manager: TIm Galligan COI'C' Image: (c) IVilliam T. IVilli",,,s. ,\rlisr, 1969 Trane, 1969 A

To Lisa, Stephen, and to the memory of Lillie

B. K.

To Suzanne D. R.H.

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PREF Linear algebra is an important course for a di verse number of students for at least two reasons. First, few subjects can claim \0 have such widespread applications in other areas of mathematics-multi variable calculus, differential equations, and probability, for example-as well as in physics, biology, chemistry. economics, fi-nance, psychology, sociology. and all fields of engineering. Second, Ihc subject presents the student at Ihc sophomore level with an excellent opportunity to learn how to handle abstract concepts.

This book provides an introduction to Ihc basic ideas and computational tech-niques of linear algebra at the sophomore level. It also includes a wide variety of carefully selected applications. These include topics of contemporary interest. such as GoogleTh1 and Global Positioning System (DPS). The book also introduces the student to working with abstract concepts. In covering the basic ideas of linear algebra. the abstract ideas are carefully balanced by considerable emphasis on the geometrical and computatio nal aspects of the subject. This edition continues to provide Ihe optional opportunity to use MATLABor other soft ware to enhance the pedagogy of the book.

What's New in the Ninth Edition We have been vcry pleased by the wide acccptance of the first eight editions of this book throughout Ihe 38 years of its life. In prcparing thi s edition. we have carefully considered many suggestions from faculty and students for improving the content and presentation of the matcrial. We have been especially gratified by hearing from the multigenerational users who used this book as students and are now using it as faculty members. Although a great many changes have been made to develop this major revision. our objective has remained the same as in the first cight editions: 10 presellf the bCl.fic ideas of linear algebra in a II/w!/Ier tlwlthe sludellf will find understandahle. To achieve this objective, thc following fcatures have been developed in this edition:

DisclLssion questions have been added to the Chapter Review material. Many of these are suitable for writ ing projects or group activities. Old Section 2.1 has been split into two sections, 2.1, Echelon Fonn of a Ma -Irix, and 2.2. Solvillg Linear SY~fellls . This will provide improved pedagogy for covering this important material. Old Chaptcr 6, Determinants, has now becomc Chapter 3, to permit earlier coverage of this material.

Old Section 3.4, Span and Linear Independence, has been split into two sec-tions, 4.3. Span, and 4.4, Linear Independence. Sincc students often have difficulties with these more abstract topics. this revision presents this material at a somewhat slO\"er pace and has more examples.

xi

xii Preface

Chapter 8, Applicatio/l.I of Eigelll'lllue.r (llId Eigelll'ecfors, is new to this edition in thi s form. It consists o f old sections 7.3, 7.6 through 7.9. material fro m old section 7.5 on the transmission of symmetric images. and o ld sections 8.1 and 8.2. More geomctric material illustrating the discussions o f diagonalization ofsym-metric matrices and singular value decompositions.

Section 1.7, Compllfer Gmphic.r, has been expanded . More applications have been added. These include networks and chemical balance equations The exposition has been expanded in many places to improve the pedagogy and more explanations have been added to show the importance of certain material and results. A simplified discussion showing how linear algebra is used in global position-ing systems (G PS) has been added. More material on recurrence relations has been added. More varied examples of vector spaces have been introduced. More material discussing the four fundamental subspaces of linear algebra have beell added.

More geometry has been added. More figures have been added.

More exercises at all levels have been added. Exercises involving real world data have been updated to include more recent data sets.

More MAT LAB exercises have been added.

EXERCISES The exercises rOml an integral part of the text. Many of them are numerical in nature, whereas others arc of a theoretical type. :-lew to this edition are Discus-sion Exercises at the end of each of the first seven chapters, which can be used for writ ing projects or group activities. Many theoretical and discussion exercises. as well as some numerical ones, call for a verbal solution. In this technological age, it is especially important to be able to write with care and precision: exercises of this type should help to sharpen this skill. Thi s edition contains almost 200 new exercises. Computer exercises. clearly indicated by a special symbol .!. arc of two types: in the fi rst eight chapters there arc exercises allowing for discovery and exploration that do not specify any particular software to be used for their . olu-tion: in Chapter tU there are 147 exercises designed to be solved using MATLAB. To extend the instructional capabilities of MAT LAB we have devcloped a set or pedagogical routines. called scripts or M-fi les, to illustrate concepts, streamline step-by-step computational procedures, and demonstrate geometric aspects of top-ics using graphical displays. We feel that MATLAB and our instructional M fil es provide an opportunity for a working pannership between the student and the com-puter that in many ways fo recasts si tuations that will occur once a student joins the technological workforce. The exercises in this chrlpter arc keyed to topics rather than individual sections o f the text. Short descripti ve headings and references to M ATI .AR commnntt., in rhnplcr 9 supply informmion nholll Ihe sel.~ of exerci.'cs

Preface xiii

The answers to all odd-numbered exercises appear in the back of the book. An In-structor's Solutions Manual (ISBN: 0-13-229655-1), containing answers to all even-numbered exercises and sol utions to all theoretical exercises, is available (to instructors only) from the publi sher.

PRESENTATION We have learned from experience that at the sophomore level, abstract ideas must be introduced quite gradually and must be based on firm foundations. Thus we begin the study of linear algebra with the treatment of matrices as mere arrays of numbers that arise naturally in the solution of sys l~ms of linear equations, a prob-lem al ready familiar to the studen1. Much al1cntion has been devoted from one edition to the next to refining and improving the pedagogical aspects of the exposi-tion. The abstract ideas are carefully balanced by the considerable emphasis on the geometrical and computational aspects of the subject. Appendix C, Illt roductioll to Proofs can be used 10 give the studcnt a quick introduction 10 the foundations of proofs in mathematics. An expanded version of this material appears in Chapter 0 of the Student Solutions Manual.

MATERIAL COVERED [n using this book, for a one-quaner linear algebra course meeti ng four times a week, no difficulty has been encountered in covering eigenvalues and eigenvectors, omil1ing thc optional matcrial. Varying the amount oftimc spent on the thcoretical material can readily change the level and pace of the course. Thus, the book can be used to teach a number of different types of courscs.

Chapter I deals v.-ith matrices and their propcnies. In this chapter we also provide an carly introduction to matrix transformations and an application of thc dot product to statistics. Methods for solving systems of lincar equations are dis-cussed in Chapter 2. Chapter 3 introduces the basic properties of determinants and some of their applications. In Chapter 4, we corne to a more abstract notion, rcal vector spaces. Here we tap some of the many geometric ideas that arise nat-urally. Thus we prove that an II -dimensional, real vector space is isomorphic to R", thc vector space of all ordered n-tuples of real numbers. or the vector space of all II x I matrices with real entries . Since R" is but a slight generalization of R2 and R3. two- and three-dimensional space are discussed at the beginning of the chapter. This shows that the notion of a finit e-dimensional. real vector space is not as remote as it may have seemed when first introduced. Chapter 5 cov-ers inner product spaces and has a strong geometric orientation. Chapter 6 deals with matrices and linear transformations: here we consider the dimension theo-rems and also appl ications to the solution of systems of linear equations. Chapter 7 considers eigenvalues and eigenvectors. In this chapter we completely solve the diagona[ization problem for symmetric matrices. Chapter 8 (optional) pre~ents an introduction to some applications of e igenvalues and eigenvectors. Section 8.3, DOll/inalll Eigellvalue and Principal Compollent Analysis, hi ghlights some very useful results in linear algebra. 11 is possible to go from Section 7.2 directly to Section 8.4. Differelllial Equations. showi ng how linear algebra is used 10 solve differcntial equations. Section 8.5. Dynamical Sy.flem.\ gives an application of lin-ear algebra to an impo