# Elementary Linear Algebra (8th Edition)

## Bernard Kolman; David R. Hill

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This book presents the basic ideas of linear algebra in a manner that users will find understandable. It offers a fine balance between abstraction/theory and computational skills, and gives readers an excellent opportunity to learn how to handle abstract concepts. Included in this comprehensive and easy-to-follow manual are these topics: linear equations and matrices; solving linear systems; real vector spaces; inner product spaces; linear transformations and matrices; determinants; eigenvalues and eigenvectors; differential equations; and MATLAB for linear algebra. Because this book gives real applications for linear algebraic basic ideas and computational techniques, it is useful as a reference work for mathematicians and those in field of computer science.

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From the Publisher:

This text has been highly successful through five previous editions for one clear reason--it presents basic ideas in a manner that students can readily understand. Coverage begins with linear systems of equations, easing students into mathematical thought processes from the outset. Kolman gradually introduces abstract ideas next, carefully supporting discussion with worked examples that illustrate the theories under review. The Sixth Edition reflects improvements in the teaching of linear algebra brought on by the calculus reform movement, as well as recommendations made by faculty and student reviewers. The result is a text that has more visualization, geometry, computation, and exercises whose solutions call for a verbal answer.

Linear algebra continues to be an important course for a diverse number of students for at least two reasons. First, few subjects can claim to have such widespread applications in other areas of mathematics—multivariable calculus, differential equations, and probability, for example—as well as in physics, biology, chemistry, economics, finance, psychology, sociology, and all fields of engineering. Second, the subject presents the student at the sophomore level with an excellent opportunity to learn how to handle abstract concepts.

This book provides an introduction to the basic ideas and computational techniques of linear algebra at the sophomore level. It includes carefully selected applications. The book introduces the student to working with abstract concepts: this includes an introduction to how to read and write proofs. In covering the basic ideas of linear algebra, the abstract ideas are carefully balanced by the considerable emphasis on the geometrical and computational aspects of the subject. This edition continues to provide the optional opportunity to use MATLAB or other software to enhance the practical side of linear algebra.

What's New in the Eighth Edition

We have been very pleased by the wide acceptance of the first seven editions of this book throughout the 34 years of its life. In preparing this edition, we have carefully considered many suggestions from faculty and students for improving the content and presentation of the material. Although a great many changes have been made to develop this major revision, our objective has remained the same as in the first seven editions: to present the basic ideas of linear algebra an a manner that the student will find understandable. To achieve this objective, the following features have been developed in this edition:

• Old Chapter 1, Linear Equations and Matrices, has been split into two chapters to improve pedagogy.
• Matrix multiplication is now covered more carefully in a separate section, Section 1.3.
• Section 1.6, Matrix Transformations, new to this edition, introduces at a very early stage some geometric applications.
• Section 1.7, Computer Graphics, has been moved from old Chapter 4 to give an application of matrix transformations.
• Several sections in old Chapters 1 and 4 have been moved to improve the organization, exposition, and flow of the material.
• Section 1.8, Correlation Coefficient, new to this edition, gives an application of the dot product to statistics.
• Section 5.6, Introduction to Homogeneous Coordinates, new to this edition, extends and generalizes earlier work on computer graphics.
• Section 7.9, Dominant Eigenvalue and Principal Component Analysis, news to this edition, includes several applications of this material. One of the applications discussed here is the way in which the highly successful search engine Google uses the dominant eigenvalue of an enormously large matrix to search the Web.
• Appendix C, Introduction to Proofs, new to this edition, provides a brief introduction to proofs in mathematics.
• The geometrical aspects of linear algebra have been greatly enhanced with 55 new figures added to this edition.
• More exercises at all levels have been added.
• Eigenvalues are now defined in terms of both real and complex numbers.
• MATLAB M-files have been upgraded to more modern versions.
• Key Terms have been added at the end of each section, reflecting the increased emphasis in mathematics on communication skills.
• A Chapter Review consisting of true/false questions and a quiz has been added to each chapter.

EXERCISES

The exercises form an integral part of the text. Many of them are numerical in nature, whereas others are of a theoretical type. The theoretical exercises (as well as many 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. This edition contains over 200 new exercises. Computer exercises, clearly indicated by a special symbol are of two types: in the first eight chapters there are exercises allowing for discovery and exploration that do not specify any particular software to be used for their solution; in Chapter 10 there are 147 exercises designed to be solved using MATLAB. To extend the instructional capabilities of MATLAB we have developed a set of pedagogical routines, called scripts or M-files, to illustrate concepts, streamline step-by-step computational procedures, and demonstrate geometric aspects of topics using graphical displays. We feel that MATLAB and our instructional M-files provide an opportunity for a working partnership between the student and the computer that in many ways forecasts situations that will occur once a student joins the technological workforce. The exercises in this chapter are keyed to topics rather than individual sections of the text. Short descriptive headings and references to MATLAB commands in Chapter 9 supply information about the sets of exercises. The answers to all odd-numbered exercises appear in the back of the book. An Instructor's Solutions Manual, containing answers to all even-numbered exercises and solutions to all theoretical exercises, is available (to instructors only) at no cost from the publisher.

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 systems of linear equations, a problem already familiar to the student. Much attention has been devoted from one edition to the next to refining and improving the pedagogical aspects of the exposition. Abstract concepts are presented along with the many computational and geometrical aspects of the subject. New to this edition is Appendix C, Introduction to Proofs, which can be used to give the student a quick introduction to the foundations of proofs in mathematics. An expanded version of this material appears in Chapter 0 of the Student Solutions Manual.

MATERIAL COVERED

In using this book, for a one-quarter linear algebra course meeting four times a week, no difficulty has been encountered in getting up to and including eigenvalues and eigenvectors, omitting the optional material. Varying the amount of time spent on the theoretical 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 courses.

Chapter 1 deals with matrices and their properties. In this chapter we also provide an early introduction to matrix transformations (setting the stage for linear transformations) and an application of the dot product to statistics. Methods for solving systems of linear equations are discussed in Chapter 2. In Chapter 3, we come to a more abstract notion, real vector spaces. Here we tap some of the many geometric ideas that arise naturally. Thus we prove that an n-dimensional, real vector space is isomorphic to Rn, the vector space of all ordered n-tuples of real numbers, or the vector space of all n x 1 matrices with real entries. Since Rn 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 finite-dimensional, real vector space is not as remote as it may have seemed when first introduced. Chapter 4 covers inner product spaces and has a strong geometric orientation. Chapter 5 deals with matrices and linear transformations; here we consider the dimension theorems and also applications to the solution of systems of linear equations. Chapter 6 introduces the basic properties of determinants and some of their applications. Chapter 7 considers eigenvalues and eigenvectors, real quadratic forms, and some applications. In this chapter we completely solve the diagonalization problem for symmetric matrices. Section 7.9, Dominant Eigenvalue and Principal Component Analysis, new to this edition, highlights some very useful results in linear algebra. Chapter 8 provides an introduction to the application of linear algebra to the solution of differential equations. It is possible to go from Section 7.2 directly to Section 8.1, showing an immediate application of the material in Section 7.2. Section 8.2, Dynamical Systems, gives an application of linear algebra to an important area of applied mathematics. Chapter 9, MATLAB for Linear Algebra, provides an introduction to MATLAB. Chapter 10, MATLAB Exercises, consists of 147 exercises that are designed to be solved using MATLAB. Appendix A reviews some very basic material dealing with sets and functions. It can be consulted at any time as needed. Appendix B introduces in a brief but thorough manner complex numbers and their use in linear algebra. Appendix C provides a brief introduction to proofs in mathematics.

MATLAB SOFTWARE

The instructional M-files that have been developed to solve the exercises in this book, in particular those in Chapter 9, are available on the following Web site: www.prenhall.com/kolman. These-M-files are designed to transform many of MATLAB'S capabilities into courseware. Although the computational exercises can be solved using a number of software packages, in our judgment MATLAB is the most suitable package for this purpose. MATLAB is a versatile and powerful software package whose cornerstone is its linear algebra capabilities. This is done LAB is the most suitable package for this purpose. MATLAB is a versatile and powerful software pack...

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