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Linear Algebra and Its Applications

David C. Lay, Steven R. Lay, and Judi J. McDonald
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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Please see our review of the fourth edition. This fifth edition is a targeted revision of the fourth, concentrating on the exercises but leaving the exposition largely unchanged. This edition adds two co-authors: Steven R. Lay (the first author’s brother, also a mathematician) and Judi J. McDonald, who co-authored the student guide for the fourth and fifth editions.

Like most mass-appeal texts these days, this one has a massive amount of web resources both for the student and the instructor. Most of these require an access key, so I did not examine them. The main difference for the fifth edition seems to be that everything is re-packaged under Pearson’s MyMathLab, but the content seems to be about the same. There is support for MATLAB, Maple, Mathematica, and even for the TI-83 (and up) calculators.

Sampling a lot of pages in the two editions it appears that no change has been made in the expository portion, except to replace some drawings and photographs. There are a large number of changes in the exercises, although nearly all of these are to change some of the numbers in the exercise without changing the solution method. I think this was done to render useless any cheat sheets for the previous edition, rather than for pedagogical reasons. The Preface (p. xxiv) says, “More than 25 percent of the exercises are new or updated, especially the computational exercises.” This figure seems accurate, but it refers almost totally to swapping one set of numbers with another equally difficult one, so the exercises, although different, are not better. The amount of new material in the exercises seems small, and the net additions to the body of the book (excluding front matter and appendices) amount to only two pages (492 vs. 494 pages total).

This is a techniques book and not a proofs book, although it does state all the important facts as theorems and gives proofs for some of them. The Preface (p. xxiii) states, “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.” I’m not sure what this means, and I did not run across any examples that seem to fit this description. The amount of added material cannot be very large, because the page count did not swell.

Bottom line: essentially the same text as before, with some different packaging for the web materials. It’s still a good text for an introductory course and is very strong on applications.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

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: 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 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: Googling 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