Linear Algebra and Its Applications [Rental Edition]

For courses in Linear Algebra.

Fosters the concepts and skills students will use in future careers

Linear Algebra and Its Applications offers a modern elementary introduction with broad, relevant applications. With  traditional texts, the early stages of the course are relatively easy as material is presented in a familiar, concrete setting; but students often hit a wall when abstract concepts are introduced. Certain concepts fundamental to the study of linear algebra (such as linear independence, vector space, and linear transformations) require time to learn–and students' understanding of them is vital. 

Lay, Lay, and McDonald 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. Throughout, the 6th Edition updates exercises, adds new applications, takes advantage of improved technology, and offers more support for conceptual learning.

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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 four university awards for teaching excellence, including, in 1996, the title of Distinguished Scholar–Teacher of the University of Maryland.  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. 

In October of 2018, David Lay passed away, 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 eight 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 three 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 and David, and their father, Dr. L. Clark Lay, are all distinguished mathematicians,  and 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 at http://mathcentral.uregina.ca/ and is a member of the Linear Algebra Curriculum Study Group.

Judi has received three teaching awards: two 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.

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 A x = 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