This book is suitable for either a sophomore-level course or for a junior/senior-level 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 sophomore-level course then one should probably spend more time on the early chapters and omit many of the sections in the later chapters. For more advanced courses one could quickly review many of the topics in the first two chapters and then do a more complete coverage of the later chapters. 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 there are computer exercises at the end of each chapter that give students the opportunity to perform numerical experiments and try to generalize the results. Applications have been scattered throughout the book. These applications can be used to motivate new material or to illustrate the relevance of material that has already been presented.
The text contains all of the topics recommended by the NSF sponsored Linear Algebra Curriculum Study Group (LACSG) and much more. Although there is more material than can be covered in a one quarter or one semester 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 of the students may use the book later as a reference and consequently may end up learning many of the omitted topics on their own.
In a later section of this preface a number of outlines are provided for one semester courses at either the sophomore-level or the junior/senior-level and with either a matrix oriented 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 an asterisk 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 one could cover the entire book in a two quarter or two semester 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, additional topics for a second course will now be made available through a special Prentice Hall web page. (See the What's New section of this preface.)
Earlier editions of this book have been used at a large number of colleges and universities for a wide variety of linear algebra courses. Thanks to the support and enthusiasm of its many users, the book is now in its fifth edition. With each new edition the book continues to evolve. Although the success of the earlier editions indicates that there is no need for fundamental changes, there are always sections and topics that can be enhanced and clarified. The author teaches two or three linear algebra classes per year and is constantly seeking better ways to present the material. Reviewers and users have also contributed many helpful suggestions. Consequently, this new edition, while retaining the essence of the previous editions, incorporates a wide array of substantive improvements.
What's New in the Fifth Edition?
The collections of applications of linear algebra has been greatly expanded for this edition. New applications cover topics from a wide variety of areas including: astronomy, ecology, chemistry, psychology, economics, signal processing, manufacturing, mechanical engineering, and statistics. Also included is a new subsection on Markov chains. Some of the new applications include:
Some of the more difficult vector spaces concepts are previewed early in the book. For example, linear combinations are covered in Chapter 1 right after the definition of matrix multiplication. This makes it easier for students when they get to the vector space chapter later. The vector space concepts are the part of the course that usually gives students the most difficulty.
There are a number of improvements in the matrix methods covered in Chapter 1. The new edition includes a more modern and improved definition of matrix multiplication, a discussion of outer product expansions, and an introduction to matrix factorizations. the LU factorization of a matrix has been moved to Chapter 1. It naturally fits in the chapter since it is a direct byproduct of Gaussian elimination whenever the process can be performed without scaling rows or interchanging the order of the rows.
The computers exercises in the book are all based on a software package called MATLAB®. These exercise sets have been expanded and updated for use with version 5.0 of MATLAB. New exercises have been added to take advantage of the improved symbolic toolbox in MATLAB 5.0. The MATLAB Appendix has also been updated and expanded. Two new sections have been added to the Appendix, one describing MATLAB's script and function files and the other describing the symbolic toolbox.
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. Since 1992 ATLAST has been offering faculty workshops using the MATLAB software package. Participants in these workshops design computer exercises, projects, and lesson plans for software-based 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 companion volume of this textbook for a nominal fee (when the two books are bundled together for class orders). More information about the ATLAST book can be obtained from the following links:
The collection of software tools (M-files) developed to accompany the ATLAST book may be downloaded from their web page:
This edition features an even greater emphasis on visualization and geometry. There are many new figures and many computer exercises and projects to help students visualize linear algebra. Furthermore the companion ATLAST book has an impression collection of software tools for visualization which are incorporated into many of its exercises and projects.
The internet provides a means for supplying additional topics requested by some users without increasing the size and cost of the book to the general audience. In particular, two new sections of supplemental material are available for this book. These new sections are not available in hard copy but are downloadable from a special page maintained by Prentice Hall. The new topics are:
Additional supplemental topics may be added in the future. Instructors adopting the book who wish to use this material should contact Prentice Hall for information on how to access the special web page.
In preparing the fifth edition, the author has carefully reviewed every section of the book. In addition to the major changes that have been listed, new exercises and examples have been added and numerous minor improvements have been made throughout the text.
Computer Exercises
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 fifty-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 contain 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. This view seems to
be gaining wide support in the greater mathematics community. The Linear
Algebra Curriculum Study Group has recommended that technology be used in
the first course in linear algebra. At meetings of all three major mathematics
societies there are now sessions devoted primarily to using computers for
teaching linear algebra. The ATLAST Project has conducted eighteen workshops
on the use of software in the teaching of linear algebra. Over four hundred
faculty members nationwide have participated in these workshops.
|
Problems or Suggestions? Order Information [ Legal Statement ] |
| ©1998 Prentice-Hall, Inc. A Simon & Schuster Company Upper Saddle River, New Jersey 07458 |