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Linear Algebra and Its Applications, 6th edition

Published by Pearson (January 24, 2020) © 2021

  • David C. Lay University of Maryland
  • Judi J. McDonald Washington State University
  • Steven R. Lay Lee University

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For courses in Linear Algebra.

Concepts and skills for future careers

Linear Algebra and Its Applications is a contemporary introduction with broad, relevant applications. Concepts such as linear independence, vector space and linear transformations require time to digest, and students' understanding of them is vital. The authors make these concepts more accessible by introducing them early in a familiar, concrete Rn setting, developing them gradually, and returning to them throughout the text so that students can grasp them when they are discussed in the abstract. The 6th Edition offers exciting new material, examples and online resources, along with new topics, vignettes and applications.

Hallmark features of this title

  • Key concepts of linear algebra are introduced within the first 7 lectures, in the concrete setting of Rn, then gradually examined from different points of view.
  • Linear transformations form a thread woven throughout, enhancing the geometric flavor of the text.
  • Eigenvalues appear fairly early, in Chapters 5 and 7.
  • Focus on visualization of concepts throughout: Each major concept is given a geometric interpretation, for more effective learning.
  • Applications are varied and relevant. Some applications appear in their own sections; others are treated within examples and exercises.
  • Meticulously constructed exercise sets in each section: An abundant supply of exercises ranges from routine computations to conceptual questions to applications.

New and updated features of this title

  • Modernized applications: After discussion with high-tech industry contacts and colleagues in applied areas, the authors have added new topics, vignettes and applications to highlight for students and faculty the linear algebraic foundational material for machine learning, artificial intelligence, data science and digital signal processing.
  • New Reasonable Answers features offer advice and exercises that help students analyze their solutions to determine if they are consistent with the data at hand and the questions asked.
  • New Projects added to the end of each chapter enhance students' exploration of various topics.
    • A list of projects appears at the end of each chapter. Projects can be used individually or in groups.
    • Topics encompass a wide range of themes, from using linear transformations to create art to exploring additional ideas in mathematics.
  • Added topics and applications prepare students with some of the foundations for machine learning, artificial intelligence, data analysis, and digital signal processing.

Features of MyLab Math for the 6th Edition

  • Free-response writing exercises have been added to MyLab 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.
  • Over 200 interactive figures enable students and instructors to play with mathematics.
    • Multiple examples can now be viewed with the click of a button, leading to the formulation of conjectures that will later be proven.
    • The geometry of linear algebra also comes alive as instructors show moving pictures illustrating concepts and definitions as they are discussed.
  1. Linear Equations in Linear Algebra
    • Introductory Example: Linear Models in Economics and Engineering
    • 1.1 Systems of Linear Equations
    • 1.2 Row Reduction and Echelon Forms
    • 1.3 Vector Equations
    • 1.4 The Matrix Equation Ax = b
    • 1.5 Solution Sets of Linear Systems
    • 1.6 Applications of Linear Systems
    • 1.7 Linear Independence
    • 1.8 Introduction to Linear Transformations
    • 1.9 The Matrix of a Linear Transformation
    • 1.10 Linear Models in Business, Science, and Engineering
    • Projects
    • Supplementary Exercises
  2. Matrix Algebra
    • Introductory Example: Computer Models in Aircraft Design
    • 2.1 Matrix Operations
    • 2.2 The Inverse of a Matrix
    • 2.3 Characterizations of Invertible Matrices
    • 2.4 Partitioned Matrices
    • 2.5 Matrix Factorizations
    • 2.6 The Leontief Input - Output Model
    • 2.7 Applications to Computer Graphics
    • 2.8 Subspaces of Rn
    • 2.9 Dimension and Rank
    • Projects 
    • Supplementary Exercises 
  3. Determinants
    • Introductory Example: Random Paths and Distortion
    • 3.1 Introduction to Determinants
    • 3.2 Properties of Determinants
    • 3.3 Cramer's Rule, Volume, and Linear Transformations
    • Projects
    • Supplementary Exercises
  4. Vector Spaces
    • Introductory Example: Space Flight and Control Systems
    • 4.1 Vector Spaces and Subspaces
    • 4.2 Null Spaces, Column Spaces, and Linear Transformations
    • 4.3 Linearly Independent Sets; Bases
    • 4.4 Coordinate Systems
    • 4.5 The Dimension of a Vector Space
    • 4.6 Change of Basis
    • 4.7 Digital Signal Processing
    • 4.8 Applications to Difference Equations
    • Projects
    • Supplementary Exercises
  5. Eigenvalues and Eigenvectors
    • Introductory Example: Dynamical Systems and Spotted Owls
    • 5.1 Eigenvectors and Eigenvalues
    • 5.2 The Characteristic Equation
    • 5.3 Diagonalization
    • 5.4 Eigenvectors and Linear Transformations
    • 5.5 Complex Eigenvalues
    • 5.6 Discrete Dynamical Systems
    • 5.7 Applications to Differential Equations
    • 5.8 Iterative Estimates for Eigenvalues
    • 5.9 Markov Chains
    • Projects
    • Supplementary Exercises
  6. Orthogonality and Least Squares
    • Introductory Example: The North American Datum and GPS Navigation
    • 6.1 Inner Product, Length, and Orthogonality
    • 6.2 Orthogonal Sets
    • 6.3 Orthogonal Projections
    • 6.4 The Gram–Schmidt Process
    • 6.5 Least-Squares Problems
    • 6.6 Machine Learning and Linear Models
    • 6.7 Inner Product Spaces
    • 6.8 Applications of Inner Product Spaces
    • Projects
    • Supplementary Exercises
  7. Symmetric Matrices and Quadratic Forms
    • Introductory Example: Multichannel Image Processing
    • 7.1 Diagonalization of Symmetric Matrices
    • 7.2 Quadratic Forms
    • 7.3 Constrained Optimization
    • 7.4 The Singular Value Decomposition
    • 7.5 Applications to Image Processing and Statistics
    • Projects
    • Supplementary Exercises
  8. The Geometry of Vector Spaces
    • Introductory Example: The Platonic Solids
    • 8.1 Affine Combinations
    • 8.2 Affine Independence
    • 8.3 Convex Combinations
    • 8.4 Hyperplanes
    • 8.5 Polytopes
    • 8.6 Curves and Surfaces
    • Projects
    • Supplementary Exercises
  9. Optimization 
    • Introductory Example: The Berlin Airlift
    • 9.1 Matrix Games
    • 9.2 Linear Programming - Geometric Method
    • 9.3 Linear Programming - Simplex Method
    • 9.4 Duality
    • Projects
    • Supplementary Exercises
  10. Finite-State Markov Chains (Online Only)
    • Introductory Example: Googling Markov Chains
    • 10.1 Introduction and Examples
    • 10.2 The Steady-State Vector and Google's PageRank
    • 10.3 Communication Classes
    • 10.4 Classification of States and Periodicity
    • 10.5 The Fundamental Matrix
    • 10.6 Markov Chains and Baseball Statistics

    Appendices

    A. Uniqueness of the Reduced Echelon Form

    B. Complex Numbers

About our authors

David C. Lay earned a B.A. from Aurora University (Illinois), and an M.A. and Ph.D. from the University of California at Los Angeles. David Lay was an educator and research mathematician for more than 40 years, mostly at the University of Maryland, College Park. He also served as a visiting professor at the University of Amsterdam, the Free University in Amsterdam, and the University of Kaiserslautern, Germany. He published more than 30 research articles on functional analysis and linear algebra.

As a founding member of the NSF-sponsored Linear Algebra Curriculum Study Group, David Lay was a leader in the current movement to modernize the linear algebra curriculum. Lay was also a coauthor of several mathematics texts, including Introduction to Functional Analysis with Angus E. Taylor, Calculus and Its Applications with L. J. Goldstein and D. I. Schneider, and Linear Algebra Gems: Assets for Undergraduate Mathematics with D. Carlson, C. R. Johnson and A. D. Porter.

David Lay received 4 university awards for teaching excellence, including the title of Distinguished Scholar-Teacher of the University of Maryland in 1996. In 1994, he was given one of the Mathematical Association of America's Awards for Distinguished College or University Teaching of Mathematics. He was elected by the university students to membership in Alpha Lambda Delta National Scholastic Honor Society and Golden Key National Honor Society. In 1989, Aurora University conferred on him the Outstanding Alumnus award. David Lay was a member of the American Mathematical Society, the Canadian Mathematical Society, the International Linear Algebra Society, the Mathematical Association of America, Sigma Xi, and the Society for Industrial and Applied Mathematics. He also served several terms on the national board of the Association of Christians in the Mathematical Sciences.

David Lay passed in October of 2018, but his legacy continues to benefit students of linear algebra as they study the subject in this widely acclaimed text.

Steven R. Lay began his teaching career at Aurora University (Illinois) in 1971, after earning an M.A. and a Ph.D. in mathematics from the University of California at Los Angeles. His career in mathematics was interrupted for 8 years while serving as a missionary in Japan. Upon his return to the States in 1998 he joined the mathematics faculty at Lee University (Tennessee) and has been there ever since. Since then he has supported his brother David in refining and expanding the scope of this popular linear algebra text, including writing most of Chapters 8 and 9. Steven is also the author of 3 college-level mathematics texts: Convex Sets and Their Applications, Analysis with an Introduction to Proof, and Principles of Algebra.

In 1985, Steven received the Excellence in Teaching Award at Aurora University. He, David and their father, Dr. L. Clark Lay, are all distinguished mathematicians; in 1989 they jointly received the Outstanding Alumnus award from their alma mater Aurora University. In 2006, Steven was honored to receive the Excellence in Scholarship Award at Lee University. He is a member of the American Mathematical Society, the Mathematics Association of America, and the Association of Christians in the Mathematical Sciences.

Judi J. McDonald became a co-author on this text's 5th Edition, having worked closely with David on the 4th Edition. She holds a B.Sc. in Mathematics from the University of Alberta, and an M.A. and Ph.D. from the University of Wisconsin. As a professor of mathematics, she has more than 40 publications in linear algebra research journals and more than 20 students have completed graduate degrees in linear algebra under her supervision. She is an associate dean of the Graduate School at Washington State University and a former chair of the Faculty Senate. She has worked with the mathematics outreach project Math Central and is a member of the Linear Algebra Curriculum Study Group.

Judi has received 3 teaching awards: 2 Inspiring Teaching awards at the University of Regina, and the Thomas Lutz College of Arts and Sciences Teaching Award at Washington State University. She also received the College of Arts and Sciences Institutional Service Award at Washington State University. Throughout her career, she has been an active member of the International Linear Algebra Society and the Association for Women in Mathematics, and has also been a member of the Canadian Mathematical Society, the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics.

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