This book presents the basic ideas of matrix and linear algebra in such a way that users from diverse backgrounds (who have had some exposure to calculus) will understand, by utilizing both algebraic and geometric reasoning. A spiral approach gradually introduces the abstract foundations of the topics involved—linear combination, closure, subspaces, linear independence/dependence, and bases. Opportunities for a variety of applications, and the optional use of MATLAB, provide hands-on explorations of computations and concepts. Chapter topics include matrices, linear systems and their solutions, Eigen information, vector spaces, inner product spaces, and linear transformations. For individuals who want to learn abstract concepts and deal with a wide variety of applications that can be drawn from fields such as physics, chemistry, biology, geology, economics, engineering, computer science, psychology, and sociology.

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PREFACE

Linear algebra is an important course for an increasing number of students in diverse disciplines. It is the course in which the majority of students have the opportunity to learn abstract concepts and deal with a wide variety of applications that can be drawn from fields such as physics, chemistry, biology, geology, economics, engineering, computer science, psychology, and sociology.

This book provides an introduction to the basic algebraic, geometric, and computational tenets of linear algebra at the sophomore level. It also stresses the abstract foundations of the topics involved, using a spiral approach. To this end we introduce many of the foundations of linear algebra by first dealing with matrices, their properties, and the algebra associated with them, while showing the geometric foundations of the topics. It is important for students to be able to associate a visual perception with certain topics, and we try to emphasize that algebra and geometry are complementary to one another. We next use a linear system of equations to develop the basic abstract notions of linear algebra. Rather than use abstract vector spaces, we concentrate on the subspaces associated with a linear system and its coefficient matrix. Chapters 1 through 4 establish these foundations and provide the opportunity to explore a variety of applications that illustrate the underlying concepts.

The second level of the spiral involves Chapters 5 through 7, in which the major ideas of beginning linear algebra are developed from a more abstract point of view. At this point the student has worked with the notions and jargon in the special cases of matrices and the subspaces associated with a linear system. Hence the reformulation of the topics to the general setting of abstract vector spaces is not as intimidating from either a conceptual or algebraic standpoint. We feel that the experience obtained in Chapters 1 through 4 will make the assimilation of ideas in the more abstract setting easier since we have established conceptual hooks for all the major concepts.

Throughout the text there as the opportunity to use MATLAB® to complement the topics. It is not required, but in today's computer-oriented society and workplace our experience is that it adds a level of understanding through hands-on involvement with computations and concepts. This certainly is in line with the Linear Algebra Curriculum Study Group recommendations and is supported by the experiences of a wide segment of mathematical educators. The idea is to use software tools within MATLAB to enhance the presentation of topics and provide opportunities for students to explore topics without the algebraic overhead involved in solving linear systems. To this end there are tools available for both the instructor and the student.

Our approach is somewhat different from that developed elsewhere. However, the major objective is to present the basic ideas of linear algebra in a manner that the student will understand by utilizing both algebraic and geometric constructs. EXERCISES

The exercises are an integral part of this text. Many of them are numerical in nature whereas others are more conceptually oriented. The conceptual exercises often cal!: for an explanation, description, or supporting reasoning to establish credence of a statement. It is extremely important in our technological age to be able to communicate with precision and to express ideas clearly. Thus exercises of this type are; designed to sharpen such skills. An instructor can choose whether to incorporate complex numbers and matrices as part of the course. While the use of MATLAB is optional, we highly encourage both instructors and students to incorporate it in some way within the instructional and study processes associated with this text.. Hence, where appropriate we have included MATLAB exercises, denoted ML exercises, in many sections. These are grouped separately from the standard exercises. If an instructor desires to use MATLAB as part of the course, then naturally it can be used with the computationally oriented standard exercises. PRESENTATION

We have learned from experience that at the sophomore level, abstract ideas must be introduced gradually and must be supported by firm reasoning. Thus we have chosen to use a spiral approach so that basic linear algebra concepts are seen first in the context of matrices and solutions of linear systems, and are then extended to a more general setting involving abstract vector spaces. Hence there is a natural grouping of the material. Chapters 1 through 4 emphasize the structure and algebra of matrices and introduce the requisite ideas to determine and analyze the solution space of linear systems. We tie the algebra to geometric constructs when possible and emphasize the interplay in these two points of view. We also provide the opportunity to use the notions of linear algebra in a variety of applications. (A list of applications appears on pages xiv-xv.) Chapters 5 through 7 extend the concepts of linear algebra to abstract vector spaces. We feel that the experiences from Chapters 1 through 4 make the assimilation of ideas in this more abstract setting easier. Instead of encountering a dual battle with abstract vector space notions and the concepts for manipulating such information, there is only the generalization to objects that are not necessarily vectors or matrices. Since we have already established the major concepts of closure, span, linear independence/dependence, subspaces, transformations, and eigenvalues/eigenvectors, the accompanying language and major manipulation steps are familiar.

We have designed features throughout the text to aid students in focusing on the topics and for reviewing the concepts developed. At the end of each section is a set of True/False Review Questions on the topics introduced. In addition, there is a list of terminology used in the section together with a set of review/discussion questions. This material can be used by the student for self study or by the instructor for writing exercises. At the end of each chapter is a Chapter Test, which provides additional practice with the main topics.

Rather than include only answers to exercises in a section at the back of the book, we have also included complete solutions to selected exercises. A number of these provide guidance on the strategy for solving a particular type of exercise, together with comments on solution procedures. Thus our Answers/Solutions to Selected Exercises incorporates features of a student study guide. The answers to all of the True/False Review Questions and the Chapter Tests are included at the back of the book. MATERIAL COVERED

Chapters 1 through 4 cover the major topics of linear algebra using only matrices and linear systems. A goal is to establish early on the language and concepts of (general) linear algebra with these simple, pervasive, and concrete constructs. Hence the order of topics is not traditional, yet carefully builds the solution space of a linear system, basic notions of linear transformations, and the eigen concepts associated with matrices. There is ample flexibility in the topics, in Chapters 3 and 4 especially, to develop a variety of paths for instruction. By careful selection of materials an instructor can also incorporate selected sections of Chapters 5 through 7, instead of using the linear ordering given in the table of contents. Varying the amount of time spent on the theoretical material can readily change the level and pace of the course.

Chapter 1 deals with matrices, their properties, and operations used to combine the information represented within this simple data structure. We use the simple versatile matrix to introduce many of the fundamental concepts of linear algebra that are later extended to more general settings. Chapter 2 develops techniques for solving linear systems of equations and for studying the properties of the sets of solutions. In particular, we investigate ways to represent the entire set of solutions of a linear system with the smallest amount of information possible. This adds an important concept to those previously developed. Chapter 3 introduces the basic properties of determinants. The concepts in this chapter can be viewed as extensions of previous material or can be treated more traditionally using recursive computation. The instructor has flexibility in choosing the extent to which the variety of material in this chapter is used. Chapter 4 considers the concepts of eigenvalues and eigenvectors of a matrix. We initially provide a geometric motivation for the ideas and follow this with an algebraic development. In this chapter we do consider that a real matrix can have complex eigenvalues and eigenvectors. We also have presented material that uses eigen information for image compression, and hence efficient transmission of information. This approach serves to illustrate geometrically the magnitude of the information that is distilled into the eigenvalues and eigenvectors. Optionally the important notion of singular value decomposition can be discussed. Chapter 5 introduces abstract vector spaces. Here we use the foundations developed for matrices and linear systems in previous chapters to extend the major concepts of linear algebra to more general settings. Chapter 6 discusses inner product spaces. The material here is again an extension of ideas previously developed for vectors in Rn (and in Cn). We revisit topics in more general settings and expand a number of applications developed earlier. We investigate the important notion of function spaces. Chapter 7 presents material on linear transformations between vector spaces that generalize the matrix transformations developed in Chapter 1. We further analyze properties and associated subspaces of linear transformations. We also extend the notion of eigenvalues and eigenvectors to this more general setting. As an application we present an introduction to fractals, which requires the use of MATLAB in order to provide a visualization component to this important area of mathematics. Chapter 8 provides a quick introduction to MATLAB and to some of the basic commands and language syntax needed to effectively use the optional ML exercises included in this text. Appendix 1 introduces in a brief but thorough manner complex numbers. Appendix 2 reviews material on summation notation. MATLAB SOFTWARE

MATLAB is a versatile and powerful software package whose cornerstone is its linear algebra capabilities. MATLAB incorporates professionally developed computer routines for linear algebra computation. The code employed by MATLAB is written in the C language and is upgraded as new versions are released. MATLAB is available from The MathWorks, Inc., 24 Prime Park Way, Natick, MA 01760, (508653-1415), e-mail info@mathworks and not distributed with this book or the instructional routines developed for the ML exercises.

The instructional M-files that have been developed to be used with the ML, exercises of this book are available from the following Prentice Hall Web site: prenhall/hillkolman. (A list of these M-files appears in Sections 8.2.) These M-files are designed to transform MATLAB's capabilities into instructional courseware. This is done by providing pedagogy that allows the student to interact with MATLAB, thereby letting the student think through the steps in the solution of a problem and relegating MATLAB to act as a powerful calculator to relieve the drudgery of tedious computation. Indeed, this is the ideal role for MATLAB in a beginning linear algebra course, for in this course, more than in many others, the tedium of lengthy computations makes it almost impossible to solve modest-sized problems. Thus by introducing pedagogy and reining in the power of MATLAB, these M-files provide a working partnership between the student and the computer. Moreover, the introduction to a powerful tool such as MATLAB early in the student's college career opens the way for other software support in higher-level courses, especially in science and engineering. SUPPLEMENTS

Instructor's Solutions Manual (0-13-019725-4): Contains answers/solutions to all exercises and is available (to instructors only) from the publisher at no cost.

Optional combination packages: Provide a MATLAB workbook at a reduced cost when packaged with this book. Any of the following three MATLAB manuals can be wrapped with this text for a small extra charge:

Hill/Zitarelli, Linear Algebra Labs with MATLAB, 2/e (0-13-505439-7) Leon/Herman/Faukenberry, ATLAST Computer Exercises for Linear Algebra (0-13270273-8) Smith, MATLAB Project Book for Linear Algebra (0-13-521337-1) ACKNOWLEDGMENTS

We benefited greatly from the class testing and useful suggestions of David E. Zitarelli of Temple University. We also benefited from the suggestions and review of the MATLAB material provided by Lila F. Roberts of Georgia Southern University.

We are pleased to express our thanks to the following reviewers: John F. Bukowski, Juniata College; Kurt Cogswell, South Dakota State University; Daniel Cunningham, Buffalo State College; Rad Dimitric, University of California, Berkeley; Daniel Kemp, South Dakota State University; Daniel King, Sarah Lawrence College; George Nakos, US Naval Academy; Brenda J. Latka, Lafayette College; Cathleen M. Zucco Teveloff, State University of New York, New Paltz; Jerome Wolbert, University of Michigan at Ann Arbor.

The numerous suggestions, comments, and criticisms of these people greatly improved this work.

We thank Dennis Kletzing, who typeset the entire manuscript.

We also thank Nina Edelman, Temple University, for critically reading page proofs. Thanks also to Blaise DeSesa for his aid in editing the exercises and solutions.

Finally, a sincere expression of thanks to Betsy Williams, George Lobell, Gale Epps, and to the entire staff at Prentice Hall for their enthusiasm, interest, and cooperation during the conception, design, production, and marketing phases of this book.

D.R.H.

B.K.

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