Elementary Linear Algebra (2nd Edition)

Book Preface

In response to concern that the first course in linear algebra was not meeting the needs of the students who take it, the Lineru· Algebra Curriculum Study Group was formed in January 1990. With the support of the National Science Foundation. it sponsored a workshop on the undergraduate linear algebra curriculum at the College of William and Mary in August of that year. Participants at the workshop included representatives from mathematics departments and other disciplines whose students study linear algebra.

Its recommendations, 1 published in January 1993, have inspired a number of textbooks, including this one. ln its core syllabus. abstract vector spaces are regarded asa supplementary topic. What dist inguishes this book from many others is its complete development of the concepts of linear algebra in R” before the introduction of abstract vector spaces. We agree wi th the following statement by the Linear Algebra Curriculum Study Group:

Furthermore. an overemphasis on abstraction may overwhelm beginning students 10 the poim where they leave the course with liu/e understanding or mastery of the basic concepts they may need in later courses and their careers. Although we believe that the first linear algebra course is a natural one in which to introduce mathematical theory and proofs, this can be accomplished without discussion
of abstract vector spaces. In addition, if abstract vector spaces are included in the first linear algebra course, we bel ieve that they can be taught more effectively after a thorough presentation of the essential topics in the more familiar context of R”. Never1heless. we have written this book so that it is possible to teach concepts in the context of abstract vector spaces immediately after the corresponding topics are discussed in R”. (Suggestions for doing so are included in the sample course descriptions on page xiv.)

Although there is no use of calculus until the introduction of vector spaces in Chapter 7, the material is aimed at students who have the mathematical maturity obtained by having taken one year of calculus. The core topics can be comfortably covered within one semester, but there is adequate material for a two-quarter course.

PEDAGOGICAL APPROACH

This text is written for a matrix-oriented course. as reconunended by the Linear Algebra CutTiculum Study Group. In our experience, such a course results in greater understanding of the concepts of linear algebra and serves the needs of students in many disciplines. It begins with the study of matrices, vectors, and systems of linear equations and gradually leads to more complicated concepts and general principles. such as linear independence, subspaces, and bases. As mentioned, this text develops all the core content of linear algebra in R” before introducing abstract vector spaces. This provides students additional opportuni ties to visualize concepts in the fami liar context of the Euclidean plane and 3-space before encountering the abstraction of vector spaces.

Our approach is based on an early introduction of the rank of a matrix. This concept is then encountered in other contexts throughout the book. For example, the rank of a matrix is used initially to check if solutions of a system of linear equations exist and are unique. Later, it is used to test if sets are linearly independent or are spanning sets for R”. Then, in Chapter 2, it is used to determine whether linear transformations are one-to-one or onto. Even though a course taught from th is book may devote less time to the study of abstract mathematics than a more traditional course. we have found that it s till serves as an excellent prerequisite to abstract algebra and an abstract second comse in linear algebra. such as one using our text Linear Algebra.

TECHNOLOGY

The Linear Algebra Study Group recommends that technology be used in the first course in linear algebra. In our experience, the use of technology, whether through computer software or supcrcalculators. greatly enhances a course taught from this book by freeing students from tedious computations and enabling them to concentrate on conceptual understanding.

Most sections include exercises designed to be worked by means of MATLAB or similar technology. Additional technology exercises are found at the end of each chapter. For MATLAB users, our website contains data files and M-files that can be downloaded. For the convenience of those wishing to use technology, we have added an appendix (Appendix D) with an introduction to MATLAB. It provides sufficient back· ground to prepare students to perform the calculations required for this book and to work the technology exercises.

EXAMPLES AND PRACTICE PROBLEMS

Our examples motivate and illustrate definitions and theoretical results. These are written to be understood by students so that an instmctor need not discuss each example in class. In fact, in our own teaching, we almost never d iscuss examples from the text, but rather present similar examples, leaving the text examples to be read by students. Many examples are accompanied by similar practice problems that enable students to test their understanding of the material in the text. Complete solutions of these practice problems are included within the text in order to help prepare students to work the exercises in each section.