Linear Algebra and Its Applications

Book Preface

The response of students and teachers to the first five editions of Linear Algebra and Its
Applications has been most gratifying. This Sixth 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  classical and leading-edge applications. The material is accessible to students with the  maturity that should come fom 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 wil I use later in their careers. The topics here fol low the recommendations of the original  Linear Algebra Curriculum Study Group (LACSG), 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. Ideas being discussed by the second Linear
Algebra Curriculum Study Group (LACSG 2.0) have also been included. 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 Sixth Edition has exciting new material, examples. and online resources. After talk­ing with high-tech industry researchers and colleagues in applied areas, we added new topics, vignettes, rmd applications with the intention of highlighting for students and faculty the linear algebraic foundational material for machine learning, artificial intelli­gence, data science, and digital signal processing.

Content Changes

• Since matrix multiplication is a highly useful skill, we added new examples in Chap­ter 2 to show how matrix multiplication is used to identify patterns and scrub data. Corresponding exercises have been created to allow students to explore using matrix multiplication in various ways.
• In our conversations with colleagues in industry and electrical engineering, we heard repeatedly how important understanding abstract vector spaces is to their work. After reading the reviewers’ comments for Chapter 4, we reorganized the chapter, condens­ing some of the material on column, row, ru1d null spaces; moving Markov chains to the end of Chapter 5; and creating a new section on signal processing. We view signals as an infinite dimensional vector space and illustrate the usefulness of linear trans­formations to filter out unwanted “vectors” (a.k.a. noise), analyze data, and enhance signals.

• By moving Markov chains to the end of Chapter 5, we can now discuss the steady state vector as an eigenvector. We also reorganized some of the summary material on determinants and change of basis to be more specific to the way they are used in this chapteL
• In Chapter 6, we present pattern recognition as an application of orthogonality, and the section on linear models now illustrates how machine learning relates to curve fitting.
• Chapter 9 on optimization was previously available only as an online file. It has now been moved into the regular textbook where it is more readily available to faculty and students. After an opening section on finding optimal strategies to two­person zero-sum games, the rest of the chapter presents an introduction to linear programming-from two-dimensional problems that can be solved geometrically to higher dimensional problems that are solved using the Simplex Method.

Other Changes

• In the high-tech industry. where most computations are done on computers, judging the validity of information and computations is an important step in preparing and r
analyzing data. In this edition, students are encouraged to lea n to analyze their own computations to see if they are consistent with the data at hand and the questions being asked. For this reason, we have added “Reasonable Answers’· advice and exercises to guide students.
• We have added a list of projects to the end of each chapter (available online at bit. ly/JOIM8gT and in My Lab Math). Some of these projects were previously r
available onJine and have a wide range of themes fom using linear transformations to create a1i to exploring additional ideas in mathematics. They can be used for group work or to enhance the learning of individual students.
• Free-response writing exercises have been added to My Lab Math, allowing faculty to ask more sophisticated questions online and create a paperless class without losing the richness of discussing how concepts relate to each other and introductory proof writing.
• The electronic interactive textbook has been changed from Wolfram CDF to Wolfram Cloud format. This allows faculty and students to interact with figures and examples on a wider variety of electronic, devices, without the need to install the CDF Player.
• PowerPoint lecture slides have been updated to cover all sections of the text and cover them more thoroughly.