This textbook is intended for a sophomore- or junior-level introductory course in linear algebra. We assume the student has had at least one course in calculus.

Philosophy of the Text

In teaching elementary linear algebra, we encountered three major problems:
    (1) Students had difficulty reading linear algebra textbooks. Frequently, they were too terse, especially where proofs of important results were concerned.

    (2) Students invariably ran into trouble as the largely computational first half of the course gave way to the more theoretical second half. Students were suddenly asked to work on a much higher level of abstraction and had difficulty with such nontrivial concepts as span, linear independence, one-to-one, onto, etc.

    (3) Most textbooks contained few, if any, guidelines about reading and writing simple mathematical proofs. However, many instructors have traditionally used a first course in linear algebra as a vehicle for familiarizing students with proof techniques, or building upon the introductory material on proofs found in an earlier course in discrete mathematics.

    This text addresses these problems. Above all, we have striven for clarity and used straightforward language throughout the book, often sacrificing brevity for clear and convincing explanation. We strongly encourage students to take advantage of the book's presentation by reading it deeply and thoroughly. We have tried to include many of the fine details that students will find useful, but which an instructor may not have time to cover in class.

    To make the transition to the second, more theoretical, half of the course easier, we have students working on proofs as quickly as possible. After a discussion of the basic properties of vectors, there is a special section (Section 1.3) on general proof techniques, with concrete examples using the material on vectors from Sections 1.1 and 1.2. The early placement of Section 1.3 helps to give students a strong foundation and build their confidence in the reading and writing of proofs. We then continue with a treatment of the fundamental properties of matrices, so that by the end of the first chapter, students have the basic algebraic tools necessary for the study of elementary linear algebra.

    We have left the proofs of some elementary theorems to the student. However, for almost every nontrivial theorem in Chapters 1 through 6, we have either included a proof, or given detailed hints which should be sufficient to enable students to provide a proof on their own. The only exception is Theorem 2.3 (uniqueness of reduced row echelon form).

    We have avoided "clever" or "sneaky" proofs, in which the last line suddenly produces "a rabbit out of a hat," because such proofs invariably frustrate students. They are given no insight into the strategy of the proof or how the deductive process was used. In fact, such proofs tend to reinforce the students' mistaken belief that they will never become competent in the art of writing proofs, and are certainly no help when students are called upon to write similar proofs of their own.

    In this text, proofs longer than one paragraph are usually written in a "top-down" manner, a concept borrowed from structured programming. A complex theorem is broken down into a secondary series of results, which together are sufficient to prove the original theorem. In this way, the student has a clear outline of the logical argument and can more easily reproduce the proof if called on to do so. This "goal-oriented" method of writing proofs is much closer to the actual deductive process that a mathematician would use in creating such a proof.
 

Major Changes for the Second Edition

    Earlier introduction of eigenvalues/eigenvectors: The material on eigenvalues and eigenvectors is now introduced much earlier in the text (Section 3.4), just before the introduction of abstract vector spaces. Students traditionally find this to be a difficult topic. By introducing eigenvalues and eigenvectors much earlier, we have the opportunity to present these concepts initially on an elementary level, and then reinforce them throughout the course with appropriate examples in subsequent sections. This "spiral approach" enables students to gain confidence with eigenvalues and eigenvectors before encountering a thorough, detailed treatment in Section 5.5.

    As a bonus, moving the material on eigenvalues/eigenvectors has allowed a more natural rearrangement of the material on orthogonality, so that all this material has now been gathered together in Chapter 6. This also gives more flexibility to the instructor in cases in which the course curriculum does not include orthogonality.

    Help with technology: Almost all students taking an introductory course in linear algebra now regularly have access to appropriate computer software or graphing calculators to reduce the amount of computational drudgery involved in a typical elementary linear algebra course. We believe that once a student has mastered the concepts of matrix multiplication and row reduction, there is no need to waste precious time in or out of class with rote computations. The use of a computational device to perform mundane tasks enables both the instructor and the student to concentrate on the theoretical ideas of the subject without becoming unduly bogged down with calculations. Therefore, we encourage the use of a computer or calculator to speed up the problem-solving process.

    While the exposition of this text does not depend on the use of any particular technology, a new appendix (Appendix D) has been added to provide a short introduction to several prominent computer packages: Maple V, Derive for Windows Version 4.10}, Mathematica 3.0, MATLAB 5 for Windows (Student Edition), and several current graphing calculators from Texas Instruments: the TI-85, TI-86, TI-89, TI-92, and TI-92 Plus. While this appendix does not give an in-depth treatment of these packages and calculators, it does illustrate how to perform several fundamental types of vector and matrix computations in each environment.

    Formal computational methods: There are now 20 computational methods presented in step-by-step form illustrating many of the fundamental processes in linear algebra. These have been placed in boxes for easier reference by the students. Several of these methods are new to this edition, including those in Section 5.3 for finding bases for the kernel and range of a linear transformation, and a method (Section 8.8) for finding the matrix of a rotation or reflection in homogeneous coordinates by using an appropriate translation to the origin. Several additional methods in numerical linear algebra are presented in Section 9.1. A chart listing all of these methods appears after this Preface. (Note: Charts have been removed from the preface for this web page.)

    Special care with abstract topics: The material in Sections 4.3 through 4.6 (Span, Linear Independence, Basis and Dimension, Constructing Special Bases) and 5.2 through 5.4 (The Matrix of a Linear Transformation, The Dimension Theorem, Isomorphism) has been carefully rewritten. These sections are the "heart" of any linear algebra text, and we have streamlined and reorganized these sections to help students focus attention on the most important concepts.

    Additional new material: The text has been enriched by over 60 new exercises, many of which have multiple parts, and many new examples. There are also two new application sections in this edition: computer graphics (Section 8.8) and least-squares solutions for inconsistent systems (Section 8.10).

    Change of emphasis: The material on elementary matrices in the first edition has been removed in order to decrease the amount of material that must be covered before encountering abstract vector spaces in Chapter 4. Elementary matrices were used in very few places in the first edition, and those sections have been rewritten to accommodate this change.

    The permutation approach to determinants has been replaced by a treatment relying on cofactor expansion. Students can now move through the material on determinants more quickly, without the need to introduce concepts from abstract algebra such as odd/even permutations.

Features

In addition to an emphasis on clarity and proof techniques, this text has the following features:

    Numerous examples and exercises: There are more than 280 numbered examples in the text, at least one for each new concept or application, to ensure that students fully understand the material before proceeding. Almost every theorem has a corresponding example to illustrate its meaning and/or usefulness.

    The text also contains an unusually large number of exercises. There are more than 750 numbered exercises, and many of these have multiple parts, for a total of more than 1750 questions. Some are purely computational. Many others ask the students to write short proofs, or to explore further consequences of the material. The exercises within each section are generally ordered by increasing difficulty, beginning with basic computational problems and moving on to more theoretical problems and proofs. Answers are provided at the end of the book for approximately half the computational exercises; these problems are marked with a star.

    Careful coverage of vector space topics: Many students have difficulties when the abstract topics of vector space, subspace, span, linear independence, and basis are introduced. These concepts represent a sharp transition from concrete problem solving to theoretical conceptualizing. To help alleviate these problems, we frequently employ a "spiral approach" in which students initially are exposed to concrete examples of an abstract concept. This helps to prepare the student for a later, more general treatment of the material. For example, students are introduced to the concept of linear combinations beginning in Section 1.1, long before linear combinations are defined for real vector spaces in Chapter 4. Similarly, the row space of a matrix is introduced in Chapter 2, thereby preparing students for the more general concepts of subspace and span in Sections 4.2 and 4.3. The methods in Section 4.6 for computing bases are adapted naturally from similar techniques discussed earlier in Chapters 2 and 4. Finally, material on complex vector spaces and more general inner product spaces is introduced in Chapter 7 only after students have mastered the corresponding concepts for real vector spaces.

    Applications: Linear algebra is a subject with a multitude of practical applications. Although there is never enough time to cover all the desired applications, we have included many standard ones so that instructors can choose their favorites. Chapter 8 is devoted entirely to applications of linear algebra, but there are also several shorter applications in Chapters 1 through 7. Instructors may choose to assign some of these applications as reading assignments outside of class. There is a chart following the Preface which lists the major linear algebra applications in this text. Another chart following the Preface lists the prerequisites required for each of the applications sections in Chapters 8 and 9. (Note: These charts are not included in this web version of the preface.)

    Subsections and summary charts: Almost every section of the text is divided into several manageable subsections to enhance clarity and readability. These subsections are individually titled to highlight the main themes of the section. Condensed versions of some useful charts are printed on the inside front and back covers for easy reference. Finally, for convenience, there is a comprehensive Symbol Table at the end of the book listing all of the major symbols employed in this text related to linear algebra and their meanings.
 

Chapter-by-Chapter Summary

    The first six chapters constitute the fundamental material covered in most elementary linear algebra courses, and this material is generalized to complex and inner product spaces in Chapter 7.

    Chapter 1 (Vectors and Matrices) introduces vectors and matrices and their fundamental operations and properties. This chapter includes a special section (Section 1.3) on proof techniques, illustrating some of the most important methods of proof and pointing out some of the pitfalls.

    Chapter 2 (Systems of Linear Equations) begins with the solution of systems of linear equations using the Gauss-Jordan Row Reduction Method. This is followed by a discussion of the uniqueness of reduced row echelon form, rank, row space, and inverses of matrices.

    Chapter 3 (Determinants and Eigenvalues) introduces the determinant (using a cofactor approach) and shows its usefulness in working with systems of linear equations. The chapter ends with an introductory treatment of eigenvalues and eigenvectors for matrices.

    Chapter 4 (Finite Dimensional Vector Spaces) begins a treatment of the abstract concepts of vector spaces and subspaces. Span, linear independence, basis and dimension, and coordinatization are covered. Several useful methods for finding bases are illustrated.

    Chapter 5 (Linear Transformations) introduces linear transformations. The matrix, kernel, and range of a linear transformation are covered. One-to-one and onto linear transformations are treated in depth, and the isomorphism of any n- dimensional real vector space with the space of real n-vectors is shown. The chapter ends with a more formal treatment of the concepts of eigenvalues and diagonalization in the context of linear transformations.

    Chapter 6 (Orthogonality) begins with orthogonal and orthonormal bases, and the Gram-Schmidt Process. Orthogonal matrices, orthogonal complements, and orthogonal projections are treated. The chapter ends with orthogonal diagonalization, a fitting culmination of the material in the first six chapters.

    Chapter 7 (Complex Vector Spaces and General Inner Products) generalizes the material of earlier chapters to complex vector spaces and general inner product spaces.

The remaining two chapters contain additional material:

    Chapter 8 (Additional Applications) is devoted to applications of linear algebra, including elementary graph theory, Ohm's Law, least-squares polynomials, Markov chains, function spaces, rotation of axes, computer graphics, differential equations, least-squares solutions, and quadratic forms.

    Chapter 9 (Numerical Methods) discusses important considerations when using a computer or calculator to perform computations in linear algebra. Numerical methods such as Gaussian elimination, the Jacobi and Gauss-Seidel methods, LDU decomposition, and the Power Method for calculating dominant eigenvalues are covered.

 There are five appendices, the fifth of which is Answers to Selected Exercises.  The others:

    Appendix A (Miscellaneous Proofs) contains proofs of five results that were omitted in the main part of the text because of length or complexity.

    Appendix B (Functions) includes a review of basic function terminology and properties, as well as a treatment of one-to-one and onto functions.

    Appendix C (Complex Numbers) contains a review of the basic properties of complex numbers.

    Appendix D (Computers and Calculators) includes a brief introduction to the use of several software packages and graphing calculators in performing basic vector and matrix operations.
 

Guide for the Instructor

    Chapters 1 through 7 have been written in a sequential fashion. Each section is generally needed as a prerequisite for what follows. Therefore, we recommend that these sections be covered in order. Three exceptions:

    Section 1.3 (An Introduction to Proofs) can be covered, in whole, or in part, at any time after Section 1.2.

    Section 3.3 (Further Properties of the Determinant) contains some material that can be omitted without affecting most of the remainder of the text. The topics of general cofactor expansion, (classical) adjoint matrix, and Cramer's Rule are used very sparingly in the rest of the text.

    Section 6.1 (Orthogonal Bases and the Gram-Schmidt Process) can be covered any time after Chapter 4, as can much of the material in Section 6.2 (Orthogonal Complements).

    Prerequisites for the material in Chapters 8 and 9 are listed in a chart following this Preface. These sections are completely independent of each other. Any section can be covered after its prerequisite has been met.

    Two suggested timetables for covering the material in this text are presented below --- one for a 3-credit course, and the other for a 4-credit course. While all the material of Chapters 1 through 6, and some of Chapter 7, would be covered in the 4-credit course, the 3-credit course would deemphasize Sections 2.3, 3.3, 5.4, 6.2, and 6.3, and would not include Chapter 7.
 
 

	3-Credit Course	4-Credit Course
Chapter 1	5 classes	6 classes
Chapter 2	4 classes	5 classes
Chapter 3	3 classes	6 classes
Chapter 4	12 classes	12 classes
Chapter 5	8 classes	9 classes
Chapter 6	2 classes	5 classes
Chapter 7		3 classes
Chapters 8/9 (selections)	2 classes	4 classes
Review	3 classes	3 classes
Tests	3 classes	3 classes
Total	42 classes	56 classes

 

Acknowledgments

    We gratefully thank all those who have helped in the publication of this book. We especially thank the acquistions editor at Academic Press, Robert Ross, who believed in our project and helped us enormously with the complexities of getting a second edition into press. We also thank Michael Sugarman, publisher at Harcourt/Academic Press, Shawn Brown, our production editor, Anja Mutic-Blessing, editorial assistant, and Kristin Landon, our copyeditor, all of whom were very helpful to us. An enormous debt of gratitude is owed to our designer and LATEX typesetters at Chrysalis Productions. Jeremy Hayhurst and his staff were helpful beyond measure and saved us countless hours in making the transition to Scientific Word.

    We also wish to thank those department chairs who strongly supported our eternal, all-consuming writing effort, especially Samuel Wiley and Charles Hofmann at La Salle University, and Agnes Rash and Jonathan Hodgson at St. Joseph's University. Agnes Rash also deserves thanks for testing an early version of the manuscript in an independent study course. We are also grateful to Charles Hofmann, who, along with Roseanne Hofmann at Montgomery County Community College, gave us valuable input regarding calculator technology.

    Paul Klingsberg has used a manuscript version of the second edition in an independent study course at St. Joseph's University, and we thank him for his comments and general helpfulness. We also thank Douglas Riddle for his sage advice to beginning authors as we wrote the first edition. We greatly appreciated the help given by the Information Technology staffs at St. Joseph's University and La Salle University, who enabled us to print beautiful copies of the manuscript at various stages. In particular, we thank Joseph Petragnani, Jeff Bachovchin, Sally Milliken, Wayne Lyle, Caroline Owens, Richard Trench, Ralph Romano, and Thomas Pasquale. Our colleague David Yang often assisted us, and we thank him as well.

    We thank those students who have classroom-tested the first edition and earlier versions of the manuscript. Their comments and suggestions have been extremely helpful, and have guided us in shaping the text in many ways.

    We acknowledge those reviewers who have given us many worthwhile suggestions. In particular, we thank the following first edition reviewers:

        C.S. Ballantine, Oregon State University
        Yuh-ching Chen, Fordham University
        Susan Jane Colley, Oberlin College
        Roland di Franco, University of the Pacific
        Colin Graham, Northwestern University
        K.G. Jinadasa, Illinois State University
        Ralph Kelsey, Denison University
        Masood Otarod, University of Scranton
        J. Bryan Sperry, Pittsburg State University
        Robert Tyler, Susquehanna University

    For reviewing the second edition, we especially thank:

    Ruth Favro, Lawrence Technological University
    Howard Hamilton, California State University
    Ray Heitmann, University of Texas, Austin
    Richard Hodel, Duke University
    James Hurley, University of Connecticut
    Jack Lawlor, University of Vermont
    Peter Nylen, Auburn University
    Ed Shea, California State University, Sacramento
 
    Last, but most important of all, we want to thank our wives, Ene and Lyn, for bearing extra burdens at home so that we could work on this text. Their love and support has been an inspiration. We also thank Ene, who conveniently works at St. Joseph's University, for ferrying various revisions of the manuscript back and forth between us. Finally, we are grateful to everyone for putting up with our sense of humor (or lack thereof) during the long years of preparation of this text.
 

A Final Note

    Transposing our original kernel of ideas into a complete linear algebra textbook took a much longer range of time than we had projected. Upon reflection, the constant deadlines functioned as determinant factors in getting the necessary work completed. Finally, after a consistent coordinatization of effort, we reached a terminal point - the first edition. Similarly, time was sparse during the complex transition to the second edition. It was a challenge to make the multiplicity of nontrivial changes that were necessary, yet still preserve the unique characteristics of the first edition. But now, at last, we have the basis to declare our linear independence from this product, which has spanned most of the last fourteen years, and ranks as an accomplishment of no minor magnitude. As an implication, we hope our lives will undergo a real transformation and become more normalized. We are now free to explore some singular new dimensions - until it is time to prepare a third (even more homogeneous) edition.