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More About This Textbook
Overview
This thorough and accessible textfrom one of the leading figures in the use of technology in linear algebragives readers a challenging and broad understanding of the subject. The author infuses key concepts with their modern practical applications to offer readers examples of how mathematics is used in the real world. This book stresses the important roles geometry and visualization play in understanding linear algebra. KEY TOPICS: Matrices and Systems of Equations, Determinants, Vector Spaces, Linear Transformations, Orthogonality, Eigenvalues, Numerical Linear Algebra MARKET: For all readers interested in linear algebra.
Editorial Reviews
From The Critics
The sixth edition of this text for a linear algebra course offers new chapter tests, earlier presentation of the singular value decomposition, new applications and examples, and improved theorem nomenclature. There is also a new subsection on outer products. New computer exercises emphasizing visualization have been added, based on the software package MATLAB. A new appendix explains basics of using the software. The text can be used for a sophomore or junior/senior course with students who are familiar with basics of differential and integral calculus. Leon teaches at the University of Massachusetts. Annotation c. Book News, Inc., Portland, OR (booknews.com)Booknews
A textbook for an undergraduate course for students familiar with the basics of differential and integral calculus from either one semester or two quarters of elementary calculus. Contains all the topics recommended by the NSF's Linear Algebra Curriculum Study Group, and more material than can be covered in a quarter or semester. Courses at the sophomore level can focus on the early chapters and omit many of the sections in later chapters. Courses at the junior or senior level can skate over the first two chapters and get to the meat. Annotation c. by Book News, Inc., Portland, Or.Booknews
Requires some familiarity with the basics of differential and integral Calculus. For 2nd, 3rd, or 4th year undergrads. Annotation c. Book News, Inc., Portland, OR (booknews.com)Product Details
Related Subjects
Meet the Author
Steven J. Leon is a Chancellor Professor of Mathematics at the University of Massachusetts Dartmouth. He has been a Visiting Professor at Stanford University, ETH Zurich (the Swiss Federal Institute of Technology), KTH (the Royal Institute of Technology in Stockholm), UC San Diego, and Brown University. His areas of specialty are linear algebra and numerical analysis.
Leon is currently serving as Chair of the Education Committee of the International Linear Algebra Society and as Contributing Editor to Image, the Bulletin of the International Linear Algebra Society. He had previously served as EditorinChief of Image from 1989 to 1997. In the 1990’s he also served as Director of the NSF sponsored ATLAST Project (Augmenting the Teaching of Linear Algebra using Software Tools). The project conducted 18 regional faculty workshops during the period from 1992–1997.
Table of Contents
Preface
What’s New in the Eighth Edition?
Computer Exercises
Overview of Text
Suggested Course Outlines
Supplementary Materials
Acknowledgments
1. Matrices and Systems of Equations
1.1 Systems of Linear Equations
1.2 Row Echelon Form
1.3 Matrix Arithmetic
1.4 Matrix Algebra
1.5 Elementary Matrices
1.6 Partitioned Matrices
Matlab Exercises
Chapter Test A
Chapter Test B
2. Determinants
2.1 The Determinant of a Matrix
2.2 Properties of Determinants
2.3 Additional Topics and Applications
Matlab Exercises
Chapter Test A
Chapter Test B
3. Vector Spaces
3.1 Definition and Examples
3.2 Subspaces
3.3 Linear Independence
3.4 Basis and Dimension
3.5 Change of Basis
3.6 Row Space and Column Space
Matlab Exercises
Chapter Test A
Chapter Test B
4. Linear Transformations
4.1 Definition and Examples
4.2 Matrix Representations of Linear Transformations
4.3 Similarity
Matlab Exercises
Chapter Test A
Chapter Test B
5. Orthogonality
5.1 The Scalar Product in R^{n}
5.2 Orthogonal Subspaces
5.3 Least Squares Problems
5.4 Inner Product Spaces
5.5 Orthonormal Sets
5.6 The Gram—Schmidt Orthogonalization Process
5.7 Orthogonal Polynomials
Matlab Exercises
Chapter Test A
Chapter Test B
6. Eigenvalues
6.1 Eigenvalues and Eigenvectors
6.2 Systems of Linear Differential Equations
6.3 Diagonalization
6.4 Hermitian Matrices
6.5 The Singular Value Decomposition
6.6 Quadratic Forms
6.7 Positive Definite Matrices
6.8 Nonnegative Matrices
Matlab Exercises
Chapter Test A
Chapter Test B
7. Numerical Linear Algebra
7.1 FloatingPoint Numbers
7.2 Gaussian Elimination
7.3 Pivoting Strategies
7.4 Matrix Norms and Condition Numbers
7.5 Orthogonal Transformations
7.6 The Eigenvalue Problem
7.7 Least Squares Problems
Matlab Exercises
Chapter Test A
Chapter Test B
Appendix: MATLAB
The MATLAB Desktop Display
Basic Data Elements
Submatrices
Generating Matrices
Matrix Arithmetic
MATLAB Functions
Programming Features
Mfiles
Relational and Logical Operators
Columnwise Array Operators
Graphics
Symbolic Toolbox
Help Facility
Conclusions
Bibliography
A. Linear Algebra and Matrix Theory
B. Applied and Numerical Linear Algebra
C. Books of Related Interest
Answers to Selected Exercises
Preface
The author is pleased to see the text reach its sixth edition. While the continued support and enthusiasm of the many users has been most gratifying, this does not mean that a mild revision is in order. Linear algebra is more exciting now than at almost any time in the past. Its applications continue to spread to more and more fields. Largely due to the computer revolution of the last half century, linear algebra has risen to a role of prominence in the mathematical curriculum rivaling that of calculus. Modern software has also made it possible to dramatically improve the way the course is taught. The author teaches linear algebra every semester and continues to seek new ways to optimize student understanding. For this edition every chapter has been carefully scrutinized and enhanced. Additionally, many of the revisions in this edition are due to the helpful suggestions received from users and reviewers. Consequently, this new edition, while retaining the essence of previous editions, incorporates a wide array of substantive improvements.
WHAT'S NEW IN THE SIXTH EDITION?New to this edition are chapter tests. At the end of each chapter there is truefalse exam testing the basic concepts covered in the chapter. Students are asked to prove or explain all of their answers.
The singular value decomposition (SVD) has emerged as one of the most important tools in matrix applications. Unfortunately, the topic is often omitted from linear algebra textbooks. When covered, it usually appears near the end of the book and classes rarely have time to get that far. To remedythis, we have moved the singular value decomposition approximately 100 pages forward in the book. It is now covered in Section 5 of Chapter 6. In this section we also show the applications of the singular value decomposition to least squares problems, principal component analysis, information retrieval, numerical rank of a matrix, and digital imaging. The SVD section nicely ties together some of the major topics, such as fundamental subspaces, orthogonality, and eigenvalues. It provides an ideal climax to a linear algebra course.
Eight applications were added to the previous edition. Some of these have been revised and improved in the current edition. A number of new applications have also been added. In Chapter 1 we show how matrices are used for search engines and information retrieval applications. This application is revisited in Chapters 5 and 6 after students have learned about orthogonality and singular values. Similarly, the statistical applications in Chapter 5 are revisited later in Chapter 6 after students have learned about the singular value decomposition.
Chapter 6 has ten new MATLAB exercises to help students to visualize eigenvalues and singular values and to help them gain geometric insight into these subjects.
Worked out examples make the textbook seem less abstract and more user friendly. Often students don't understand what a theorem says until they see a worked out example that illustrates the theorem. The impressive collection of examples was often cited as one of the strong points of the first edition of this book. This collection has continued to grow and improve with each new edition. More examples have been added throughout the sixth edition, and many of the previous examples lave been revised and improved. Now, for example, the numbered of worked out examples in Chapter 1 has increased from 32 to 34. In a number of cases color shading is now used to emphasize how rows and columns are paired off in matrix computations.
Throughout this edition we have made a special effort to assign names to theorems so as to emphasize the importance of the results. Also, it is easier to refer back to a theorem if it has a name. We have added a new theorem to Chapter 6. This theorem does have a name, The Principal Axes Theorem.
In Chapter 5 the order of two of the sections has been reversed. Least squares problems are now covered before the section on general inner product spaces. To facilitate this change, some new material was added to Section 1 of the chapter. With this new ordering it is possible for classes that only treat Euclidean vector spaces to skip most of Section 4. These classes need only introduce the inner product notation in Section 4 and then move on to the next section or, if pressed for time, skip ahead to the next chapter.
A new subsection on outer product expansions has been added to Chapter 1. Outer product expansions are used in later chapters applications such as digital imaging.
Prentice Hall has developed a special Web site to accompany this book. This site includes a host of materials for both students and instructors. The Web pages are being extensively revised for the sixth edition and an exciting collection of new interactive course materials is currently being developed as we go to press. Some of the other features to be included on the Web pages are a collection of links with downloadable materials relating to each of the chapters in the book and a collection of application projects that are related to the topics covered in the book. You can also download two supplemental chapters for this book from the Prentice Hall site. The new chapters are:
ATLAST (Augmenting the Teaching of Linear Algebra through the use of Software Tools) is an NSF sponsored project to encourage and facilitate the use of software in the teaching of linear algebra. During a five year period, 19921997, the ATLAST Project conducted 18 faculty workshops using the MATLAB software package. Participants in these workshops designed computer exercises, projects, and lesson plans for softwarebased teaching of linear algebra. A selection of these materials has been published as a manual. ATLAST Computer Exercises for Linear Algebra (Prentice Hall, 1997). The ATLAST book is available as a free companion volume to this textbook when the two books are wrapped together for class orders. The ISBN for ordering the twobook bundle is given in the Supplementary Materials section of this Preface. The collection of software tools (Mfiles) developed to accompany the ATLAST book may be downloaded from the ATLAST Web site. You can link to the ATLAST site from the Prentice Hall Web page for this book. A second edition of the ATLAST book is in preparation and publication is expected in the fall of 2002. New developments related to the ATLAST Project and manual will be posted on the ATLAST Web site.
A collection of ATLAST Mathematica Notebooks has been developed by Richard Neidinger of Davidson College. The collection contains Mathematical versions of the ATLAST projects and exercises. It can be downloaded for free from the ATLAST Web pages.
A new manual, Visualizing Linear Algebra Using Maple, by Sandra Keith, is now available as a companion volume to this book. The manual provides an ideal vehicle for those wishing to teach the course using Maple. The Keith manual is offered as a bundle with this book at a special discount price. The ISBN for ordering the twobook bundle is given in the Supplementary Materials section of this Preface.
A new manual, Understanding Linear Algebra Using MATLAB, by Irwin and Margaret Kleinfeld, is now available as a companion volume to this book. The book has MATLAB problems and projects suitable for a first course in linear algebra. The manual is offered as a bundle with this book at a special discount price. The ISBN for ordering the twobook bundle is given in the Supplementary Materials section of this Preface.
A new student study guide has been developed to accompany this edition. The guide is described in the Supplementary Materials section of this preface.
In preparing the sixth edition, the author has carefully reviewed every section of the book. In addition to the major changes that have been listed, many new exercises have been added and numerous minor improvements have been made throughout the text.
This edition contains a section of computing exercises at the end of each chapter. These exercises are based on the software package MATLAB. The MATLAB Appendix in the book explains the basics of using the software. MATLAB has the advantage that it is a powerful tool for matrix computations and yet it is easy to learn. After reading the Appendix, students should be able to do the computing exercises without having to refer to any other software books or manuals. To help students get started we recommend one 50 minute classroom demonstration of the software. The assignments can be done either as ordinary homework assignments or as part of a formally scheduled computer laboratory course.
As mentioned previously, the ATLAST book is available as a companion volume to supplement the computer exercises in this book. Each of the eight chapters of the ATLAST book contains a section of short exercises and a section of longer projects.
While the course can be taught without any reference to the computer, we believe that computer exercises can greatly enhance student learning and provide a new dimension to linear algebra education. The Linear Algebra Curriculum Study Group has recommended that technology be used for a first course in linear algebra, and this view is generally accepted throughout the greater mathematics community.
OVERVIEW OF TEXTThis book is suitable for either a sophomorelevel course or for a junior/seniorlevel course. The student should have some familiarity with the basics of differential and integral calculus. This prerequisite can be met by either one semester or two quarters of elementary calculus.
If the text is used for a sophomorelevel course, the instructor should probably spend more time on the early chapters and omit many of the sections in the later chapters. For more advanced courses a quick review of many of the topics in the first two chapters and then a more complete coverage of the later chapters would be appropriate. The explanations in the text are given in sufficient detail so that beginning students should have little trouble reading and understanding the material. To further aid the student, a large number of examples have been worked out completely. Additionally, computer exercises at the end of each chapter give students the opportunity to perform numerical experiments and try to generalize the results. Applications are presented throughout the book. These applications can be used to motivate new material or to illustrate the relevance of material that has already been covered.
The text contains all the topics recommended by the National Science Foundation (NSF) sponsored Linear Algebra Curriculum Study Group (LACSG) and much more. Although there is more material than can be covered in a onequarter or onesemester course, it is the author's feeling that it is easier for an instructor to leave out or skip material than it is to supplement a book with outside material. Even if many topics are omitted, the book should still provide students with a feeling for the overall scope of the subject matter. Furthermore, many students may use the book later as a reference and consequently may end up learning many of the omitted topics on their own.
In the next section of this preface a number of outlines are provided for onesemester courses at either the sophomore level or the junior/senior level and with either a matrixoriented emphasis or a slightly more theoretical emphasis. To further aid the instructor in the choice of topics, three sections have been designated as optional and are marked with a dagger in the table of contents. These sections are not prerequisites for any of the following sections in the book. They may be skipped without any loss of continuity.
Ideally the entire book could be covered in a twoquarter or twosemester sequence. Although two semesters of linear algebra has been recommended by the LACSG, it is still not practical at many universities and colleges. At present there is no universal agreement on a core syllabus for a second course. Indeed, if all of the topics that instructors would like to see in a second course were included in a single volume, it would be a weighty (and expensive) book. An effort has been made in this text to cover all of the basic linear algebra topics that are necessary for modern applications. Furthermore, two additional chapters for a second course are available for downloading from the Internet. See the special Prentice Hall Web page discussed earlier.
SUPPLEMENTARY MATERIALS