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Howard Anton, Chris Rorres Elementary Linear Algebra PDF

1226 Pages·2008·19.7 MB·English
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Preview Howard Anton, Chris Rorres Elementary Linear Algebra

P R E F A C E This textbook is an expanded version of Elementary Linear Algebra, Ninth Edition, by Howard Anton. The first ten chapters of this book are identical to the first ten chapters of that text; the eleventh chapter consists of 21 applications of linear algebra drawn from business, economics, engineering, physics, computer science, approximation theory, ecology, sociology, demography, and genetics. The applications are, with one exception, independent of one another and each comes with a list of mathematical prerequisites. Thus, each instructor has the flexibility to choose those applications that are suitable for his or her students and to incorporate each application anywhere in the course after the mathematical prerequisites have been satisfied. This edition of Elementary Linear Algebra, like those that have preceded it, gives an elementary treatment of linear algebra that is suitable for students in their freshman or sophomore year. The aim is to present the fundamentals of linear algebra in the clearest possibleway; pedagogy is the main consideration. Calculus is not a prerequisite, but there are clearly labeled exercises and examples for students who have studied calculus. Those exercises can be omitted without loss of continuity. Technology is also not required, but for those who would like to use MATLAB, Maple, Mathematica, or calculators with linear algebra capabilities, exercises have been included at the ends of the chapters that allow for further exploration of that chapter's contents. SUMMARY OF CHANGES IN THIS EDITION This edition contains organizational changes and additional material suggested by users of the text. Most of the text is unchanged. The entire text has been reviewed for accuracy, typographical errors, and areas where the exposition could be improved or additional examples are needed. The following changes have been made: Section 6.5 has been split into two sections: Section 6.5 Change of Basis and Section 6.6 Orthogonal Matrices. This allows for sharper focus on each topic. A new Section 4.4 Spaces of Polynomials has been added to further smooth the transition to general linear transformations, and a new Section 8.6 Isomorphisms has been added to provide explicit coverage of this topic. Chapter 2 has been reorganized by switching Section 2.1 with Section 2.4. The cofactor expansion approach to determinants is now covered first and the combinatorial approach is now at the end of the chapter. Additional exercises, including Discussion and Discovery, Supplementary, and Technology exercises, have been added throughout the text. In response to instructors' requests, the number of exercises that have answers in the back of the book has been reduced considerably. The page design has been modified to enhance the readability of the text. A new section on the earliest applications of linear algebra has been added to Chapter 11. This section shows how linear equations were used to solve practical problems in ancient Egypt, Babylonia, Greece, China, and India. Hallmark Features Relationships Between Concepts One of the important goals of a course in linear algebra is to establish the intricate thread of relationships between systems of linear equations, matrices, determinants, vectors, linear transformations, and eigenvalues. That thread of relationships is developed through the following crescendo of theorems that link each new idea with ideas that preceded it: 1.5.3, 1.6.4, 2.3.6, 4.3.4, 5.6.9, 6.2.7, 6.4.5, 7.1.5. These theorems bring a coherence to the linear algebra landscape and also serve as a constant source of review. Smooth Transition to Abstraction The transition from to general vector spaces is often difficult for students. To smooth out that transition, the underlying geometry of is emphasized and key ideas are developed in before proceeding to general vector spaces. Early Exposure to Linear Transformations and Eigenvalues To ensure that the material on linear transformations and eigenvalues does not get lost at the end of the course, some of the basic concepts relating to those topics are developed early in the text and then reviewed and expanded on when the topic is treated in more depth later in the text. For example, characteristic equations are discussed briefly in the chapter on determinants, and linear transformations from to are discussed immediately after is introduced, then reviewed later in the context of general linear transformations. About the Exercises Each section exercise set begins with routine drill problems, progresses to problems with more substance, and concludes with theoretical problems. In most sections, the main part of the exercise set is followed by the Discussion and Discovery problems described above. 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. The technology exercises follow the supplementary exercises and are classified according to the section in which we suggest that they be assigned. Data for these exercises in MATLAB, Maple, and Mathematica formats can be downloaded from www.wiley.com/college/anton. About Chapter 11 This chapter consists of 21 applications of linear algebra. With one clearly marked exception, each application is in its own independent section, so that sections can be deleted or permuted freely to fit individual needs and interests. Each topic begins with a list of linear algebra prerequisites so that a reader can tell in advance if he or she has sufficient background to read the section. Because the topics vary considerably in difficulty, we have included a subjective rating of each topic—easy, moderate, more difficult. (See “A Guide for the Instructor” following this preface.) Our evaluation is based more on the intrinsic difficulty of the material rather than the number of prerequisites; thus, a topic requiring fewer mathematical prerequisites may be rated harder than one requiring more prerequisites. Because our primary objective is to present applications of linear algebra, proofs are often omitted. We assume that the reader has met the linear algebra prerequisites and whenever results from other fields are needed, they are stated precisely (with motivation where possible), but usually without proof. Since there is more material in this book than can be covered in a one-semester or one-quarter course, the instructor will have to make a selection of topics. Help in making this selection is provided in the Guide for the Instructor below. Supplementary Materials for Students Student Solutions Manual, Ninth Edition—This supplement provides detailed solutions to most theoretical exercises and to at least one nonroutine exercise of every type. (ISBN 0-471-43329-2) Data for Technology Exercises is provided in MATLAB, Maple, and Mathematica formats. This data can be downloaded from www.wiley.com/college/anton. Linear Algebra Solutions—Powered by JustAsk! invites you to be a part of the solution as it walks you step-by-step through a total of over 150 problems that correlate to chapter materials to help you master key ideas. The powerful online problem-solving tool provides you with more than just the answers. Supplementary Materials for Instructors Instructor's Solutions Manual—This new supplement provides solutions to all exercises in the text. (ISBN 0-471-44798-6) Test Bank—This includes approximately 50 free-form questions, five essay questions for each chapter, and a sample cumulative final examination. (ISBN 0-471-44797-8) eGrade—eGrade is an online assessment system that contains a large bank of skill-building problems, homework problems, and solutions. Instructors can automate the process of assigning, delivering, grading, and routing all kinds of homework, quizzes, and tests while providing students with immediate scoring and feedback on their work. Wiley eGrade “does the math”… and much more. For more information, visit http://www.wiley.com/college/egrade or contact your Wiley representative. Web Resources—More information about this text and its resources can be obtained from your Wiley representative or from www.wiley.com/college/anton. A GUIDE FOR THE INSTRUCTOR Linear algebra courses vary widely between institutions in content and philosophy, but most courses fall into two categories: those with about 35–40 lectures (excluding tests and reviews) and those with about 25–30 lectures (excluding tests and reviews). Accordingly, I have created long and short templates as possible starting points for constructing a course outline. In the long template I have assumed that all sections in the indicated chapters are covered, and in the short template I have assumed that instructors will make selections from the chapters to fit the available time. Of course, these are just guides and you may want to customize them to fit your local interests and requirements. The organization of the text has been carefully designed to make life easier for instructors working under time constraints: A brief introduction to eigenvalues and eigenvectors occurs in Sections 2.3 and 4.3, and linear transformations from to are discussed in Chapter 4. This makes it possible for all instructors to cover these topics at a basic level when the time available for their more extensive coverage in Chapters 7 and 8 is limited. Also, note that Chapter 3 can be omitted without loss of continuity for students who are already familiar with the material. Long Template Short Template Chapter 1 7 lectures 6 lectures Chapter 2 4 lectures 3 lectures Chapter 4 4 lectures 4 lectures Chapter 5 7 lectures 6 lectures Chapter 6 6 lectures 3 lectures Long Template Short Template Chapter 7 4 lectures 3 lectures Chapter 8 6 lectures 2 lectures Total 38 lectures 27 lectures Variations in the Standard Course Many variations in the long template are possible. For example, one might create an alternative long template by following the time allocations in the short template and devoting the remaining 11 lectures to some of the topics in Chapters 9, 10 and 11. An Applications-Oriented Course Once the necessary core material is covered, the instructor can choose applications from Chapter 9 or Chapter 11. The following table classifies each of the 21 sections in Chapter 11 according to difficulty: Easy. The average student who has met the stated prerequisites should be able to read the material with no help from the instructor. Moderate. The average student who has met the stated prerequisites may require a little help from the instructor. More Difficult. The average student who has met the stated prerequisites will probably need help from the instructor. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 EASY • • • MODERATE • • • • • • • • • • • MORE • • • • • • • DIFFICULT Copyright © 2005 John Wiley & Sons, Inc. All rights reserved. A C K N O W L E D G E M E N T S We express our appreciation for the helpful guidance provided by the following people: REVIEWERS AND CONTRIBUTORS Marie Aratari, Oakland Community College Nancy Childress, Arizona State University Nancy Clarke, Acadia University Aimee Ellington, Virginia Commonwealth University William Greenberg, Virginia Tech Molly Gregas, Finger Lakes Community College Conrad Hewitt, St. Jerome's University Sasho Kalajdzievski, University of Manitoba Gregory Lewis, University of Ontario Institute of Technology Sharon O'Donnell, Chicago State University Mazi Shirvani, University of Alberta Roxana Smarandache, San Diego State University Edward Smerek, Hiram College Earl Taft, Rutgers University AngelaWalters, Capitol College Mathematical Advisors Special thanks are due to two very talented mathematicians who read the manuscript in detail for technical accuracy and provided excellent advice on numerous pedagogical and mathematical matters. Philip Riley, James Madison University Laura Taalman, James Madison University Special Contributions The talents and dedication of many individuals are required to produce a book such as the one you now hold in your hands. The following people deserve special mention: Jeffery J. Leader–for his outstanding work overseeing the implementation of numerous recommendations and improvements in this edition. Chris Black, Ralph P. Grimaldi, and Marie Vanisko–for evaluating the exercise sets and making helpful recommendations. Laurie Rosatone–for the consistent and enthusiastic support and direction she has provided this project. Jennifer Battista–for the innumerable things she has done to make this edition a reality. Anne Scanlan-Rohrer–for her essential role in overseeing day-to-day details of the editing stage of this project. Kelly Boyle and Stacy French–for their assistance in obtaining pre-revision reviews. Ken Santor–for his attention to detail and his superb job in managing this project. Techsetters, Inc.–for once again providing beautiful typesetting and careful attention to detail. Dawn Stanley–for a beautiful design and cover. The Wiley Production Staff–with special thanks to Lucille Buonocore, Maddy Lesure, Sigmund Malinowski, and Ann Berlin for their efforts behind the scenes and for their support on many books over the years. HOWARD ANTON CHRIS RORRES Copyright © 2005 John Wiley & Sons, Inc. All rights reserved. 1 C H A P T E R Systems of Linear Equations and Matrices I NTRODUCTION: Information in science and mathematics is often organized into rows and columns to form rectangular arrays, called “matrices” (plural of “matrix”). Matrices are often tables of numerical data that arise from physical observations, but they also occur in various mathematical contexts. For example, we shall see in this chapter that to solve a system of equations such as all of the information required for the solution is embodied in the matrix and that the solution can be obtained by performing appropriate operations on this matrix. This is particularly important in developing computer programs to solve systems of linear equations because computers are well suited for manipulating arrays of numerical information. However, matrices are not simply a notational tool for solving systems of equations; they can be viewed as mathematical objects in their own right, and there is a rich and important theory associated with them that has a wide variety of applications. In this chapter we will begin the study of matrices. Copyright © 2005 John Wiley & Sons, Inc. All rights reserved. 1.1 Systems of linear algebraic equations and their solutions constitute one of the INTRODUCTION TO major topics studied in the course known as “linear algebra.” In this first section we shall introduce some basic terminology and discuss a method for solving such SYSTEMS OF LINEAR systems. EQUATIONS Linear Equations Any straight line in the -plane can be represented algebraically by an equation of the form where , , and b are real constants and and are not both zero. An equation of this form is called a linear equation in the variables x and y. More generally, we define a linear equation in the n variables , , …, to be one that can be expressed in the form where , , …, , and b are real constants. The variables in a linear equation are sometimes called unknowns. EXAMPLE 1 Linear Equations The equations are linear. Observe that a linear equation does not involve any products or roots of variables. All variables occur only to the first power and do not appear as arguments for trigonometric, logarithmic, or exponential functions. The equations are not linear. A solution of a linear equation is a sequence of n numbers , , …, such that the equation is satisfied when we substitute , , …, . The set of all solutions of the equation is called its solution set or sometimes the general solution of the equation. EXAMPLE 2 Finding a Solution Set Find the solution set of (a) , and (b) . Solution (a) To find solutions of (a), we can assign an arbitrary value to x and solve for y, or choose an arbitrary value for y and solve for x. If we follow the first approach and assign x an arbitrary value t, we obtain These formulas describe the solution set in terms of an arbitrary number t, called a parameter. Particular numerical solutions can be obtained by substituting specific values for t. For example, yields the solution , ; and yields the solution , . If we follow the second approach and assign y the arbitrary value t, we obtain Although these formulas are different from those obtained above, they yield the same solution set as t varies over all possible real numbers. For example, the previous formulas gave the solution , when , whereas the formulas immediately above yield that solution when . Solution (b) To find the solution set of (b), we can assign arbitrary values to any two variables and solve for the third variable. In particular, if we assign arbitrary values s and t to and , respectively, and solve for , we obtain Linear Systems A finite set of linear equations in the variables , , …, is called a system of linear equations or a linear system. A sequence of numbers , , …, is called a solution of the system if , , …, is a solution of every equation in the system. For example, the system has the solution , , since these values satisfy both equations. However, , , is not a solution since these values satisfy only the first equation in the system. Not all systems of linear equations have solutions. For example, if we multiply the second equation of the system by , it becomes evident that there are no solutions since the resulting equivalent system has contradictory equations. A system of equations that has no solutions is said to be inconsistent; if there is at least one solution of the system, it is called consistent. To illustrate the possibilities that can occur in solving systems of linear equations, consider a general system of two linear equations in the unknowns x and y: The graphs of these equations are lines; call them and . Since a point (x, y) lies on a line if and only if the numbers x and y satisfy the equation of the line, the solutions of the system of equations correspond to points of intersection of and . There are three possibilities, illustrated in Figure 1.1.1:

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This edition of Elementary Linear Algebra, like those that have preceded it, A new section on the earliest applications of linear algebra has been added to Student Solutions Manual, Ninth Edition—This supplement provides
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