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Linear Algebra and Its Applications

David C. Lay
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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This is a no-frills textbook for a one-semester course in linear algebra that focuses very heavily on algorithms and applications. Despite the no-frills approach, it is still a long book and it covers everything that would normally be in an introductory course, and a lot that would not be. There is much more material here than could be covered in a semester.

The breadth of applications is especially impressive: although most of them use extremely simplified models of the thing being studied, they really do give you a good understanding of how linear algebra is used in practice. The applications cover many areas of science, business, and engineering, with a lot of dynamical systems examples. There are also many notes on numerical considerations. None of this goes into enough depth to make you an expert (that would be impossible in a one-semester introductory course), or even able to tackle such applications on your own, but it does give you a good understanding of how linear algebra is used and why it is important.

Very Good Feature: a short set of Practice Problems before each problem set. The answers in the back of the book are the typical short phrases or final answers, but the Practice Problems write out a complete solution, in complete sentences, as we hope students will do.

This is primarily a linear-equations book, not a linear-spaces book or even a matrices book. The book gradually introduces more abstract structures as it goes along, including linear transformations, vector spaces, orthogonality, and n-dimensional geometry. I thought this was not entirely successful: the abstractions seem not to lead anywhere, unlike the algorithms that are always developed with an eye to applications. Technology is not treated in the narrative, but many of the exercises have a technology component, and data sets and code are available on the web. The examples in the text are chosen so they have neat solutions.

The chapter on eigenvalues is relatively weak; it covers all the techniques, but the applications are limited to dynamical systems and the interest is in taking high powers of a matrix. The book is unusual in having a lengthy discussion of least-squares methods, brought in specifically because in real problems your system may be inconsistent and you need some way to develop a useful answer.

Given the practical emphasis, it is anomalous that a whole chapter of 26 pages is devoted to determinants. The book gives frequent warnings that the material is most useful in theoretical questions, but it is included anyway. This is another topic that doesn’t seem to go anywhere. The only justification given is that it allows us to determine when a matrix is invertible, but we already learned how to do that by row reduction in the first chapter.

There is a printed student guide available and a great deal of supplemental material for students on the web, including two additional chapters intended for a second course. I did not examine any of this material.

A somewhat similar book is Gilbert Strang’s Introduction to Linear Algebra. The applications are much weaker in Strang, but the linear-spaces material is much better integrated. Your knowledge of linear algebra as a subject will be much stronger from reading Strang than from reading Lay. Strang is more fun to read, too: Strang really loves linear algebra and it shows, while Lay is very dry (except for one isolated joke: a sight gag in the discussion of linear transformations involving a sheared sheep).

Bottom line: Very well-done introduction, despite several flaws, and especially strong on applications. It is a text that most students would get value from.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

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

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

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

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 Rank

4.7 Change of Basis

4.8 Applications to Difference Equations

4.9 Applications to Markov Chains

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

Supplementary Exercises


6. Orthogonality and Least Squares

Introductory Example: Readjusting the North American Datum

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 Applications to Linear Models

6.7 Inner Product Spaces

6.8 Applications of Inner Product Spaces

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

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


9. Optimization (Online Only)

Introductory Example: The Berlin Airlift

9.1 Matrix Games

9.2 Linear Programming–Geometric Method

9.3 Linear Programming–Simplex Method

9.4 Duality


10. Finite-State Markov Chains (Online Only)

Introductory Example: Google and Markov Chains

10.1 Introduction and Examples

10.2 The Steady-State Vector and Google's PageRank

10.3 Finite-State Markov Chains

10.4 Classification of States and Periodicity

10.5 The Fundamental Matrix

10.6 Markov Chains and Baseball Statistics



A. Uniqueness of the Reduced Echelon Form

B. Complex Numbers