Linear Algebra and Its Applications (5th Edition)

Book Preface

The response of students and teachers to the first four editions of Linear Algebra and Its Applications has been most gratifying. This Fifth Edition provides substantial support both for teaching and for using technology in the course. As before, the text provides a modern elementary introduction to linear algebra and a broad selection of interesting applications. The material is accessible to students with the maturity that should come from successful completion of two semesters of college-level mathematics, usually calculus.

The main goal of the text is to help students master the basic concepts and skills they will use later in their careers. The topics here follow the recommendations of the Linear Algebra Curriculum Study Group, which were based on a careful investigation of the real needs of the students and a consensus among professionals in many disciplines that use linear algebra. We hope this course will be one of the most useful and interesting mathematics classes taken by undergraduates.

WHAT’S NEW IN THIS EDITION

The main goals of this revision were to update the exercises, take advantage of improvements in technology, and provide more support for conceptual learning.
1. Support for the Fifth Edition is offered through MyMathLab. MyMathLab, from Pearson, is the world’s leading online resource in mathematics, integrating interactive homework, assessment, and media in a flexible, easy-to-use format. Students submit homework online for instantaneous feedback, support, and assessment. This system works particularly well for computation-based skills. Many additional resources are also provided through the MyMathLab web site.

2. The Fifth Edition of the text is available in an interactive electronic format. Using the CDF player, a free Mathematica player available from Wolfram, students can interact with figures and experiment with matrices by looking at numerous examples with just the click of a button. The geometry of linear algebra comes alive through these interactive figures. Students are encouraged to develop conjectures through experimentation and then verify that their observations are correct by examining the relevant theorems and their proofs. The resources in the interactive version of the text give students the opportunity to play with mathematical objects and ideas much as we do with our own research. Files for Wolfram CDF Player are also available for classroom presentations.

3. The Fifth Edition includes additional support for concept- and proof-based learning. Conceptual Practice Problems and their solutions have been added so that most sections now have a proof- or concept-based example for students to review. Additional guidance has also been added to some of the proofs of theorems in the body of the textbook.

4. More than 25 percent of the exercises are new or updated, especially the computational exercises. The exercise sets remain one of the most important features of this book, and these new exercises follow the same high standard of the exercise sets from the past four editions. They are crafted in a way that reflects the substance of each of the sections they follow, developing the students’ confidence while challenging them to practice and generalize the new ideas they have encountered.

DISTINCTIVE FEATURES

Early Introduction of Key Concepts
Many fundamental ideas of linear algebra are introduced within the first seven lectures, in the concrete setting of Rn, and then gradually examined from different points of view. Later generalizations of these concepts appear as natural extensions of familiar ideas, visualized through the geometric intuition developed in Chapter 1. A major achievement of this text is that the level of difficulty is fairly even throughout the course.

A Modern View of Matrix Multiplication
Good notation is crucial, and the text reflects the way scientists and engineers actually use linear algebra in practice. The definitions and proofs focus on the columns of a matrix rather than on the matrix entries. A central theme is to view a matrix–vector product Ax as a linear combination of the columns of A. This modern approach simplifies many arguments, and it ties vector space ideas into the study of linear systems.

Linear Transformations
Linear transformations form a “thread” that is woven into the fabric of the text. Their use enhances the geometric flavor of the text. In Chapter 1, for instance, linear transformations provide a dynamic and graphical view of matrix–vector multiplication.

Eigenvalues and Dynamical Systems
Eigenvalues appear fairly early in the text, in Chapters 5 and 7. Because this material is spread over several weeks, students have more time than usual to absorb and review these critical concepts. Eigenvalues are motivated by and applied to discrete and continuous dynamical systems, which appear in Sections 1.10, 4.8, and 4.9, and in five sections of Chapter 5. Some courses reach Chapter 5 after about five weeks by covering Sections 2.8 and 2.9 instead of Chapter 4. These two optional sections present all the vector space concepts from Chapter 4 needed for Chapter 5.

Orthogonality and Least-Squares Problems
These topics receive a more comprehensive treatment than is commonly found in beginning texts. The Linear Algebra Curriculum Study Group has emphasized the need for a substantial unit on orthogonality and least-squares problems, because orthogonality plays such an important role in computer calculations and numerical linear algebra and because inconsistent linear systems arise so often in practical work.