This is a conventional and overpriced introduction to linear algebra that aims for breadth rather than depth. There is more than enough material here for a one-year course, arranged with vectors, matrices, systems of equations, and eigenvalues towards the beginning, and slightly more abstract treatments of orthogonality, spaces, and metrics following. The book touches on an amazing variety of topics, including things rarely seen in introductory texts such as Gerschgorin disks for eigenvalues. Its only conspicuous omission is linear programming. Numerical methods are not addressed explicitly, but most of the background such as Gaussian elimination and LU-decomposition is covered. The author says in the Preface (p. x) that “I also believe strongly that linear algebra is essentially about vectors”, and while the book does not literally spend most of its time on vectors, it does make heavy use of geometry and pictures.
The treatment often feels shallow, touching on a subject just long enough for the student to have some familiarity with it. As an example, what is described as a “crash course” in determinants takes 29 pages, and that only covers expansion by minors, Cramer’s rule, and row and column operations. This is a 720-page book, but it is long not because it is wordy but because it has so many examples. My rough estimate is that about 40% of the book is narrative and the other 60% is worked examples and exercises. Overall the exercises are quite good, with the minority being devoted to drill and the majority asking for some kind of reasoning or logical argument. The book includes a number of proofs, both in the body and in the exercises, but does not emphasize proving. The book includes a long list of “applications”, but most of these only sketch very briefly ways that linear algebra can be used, and there’s not enough information to teach you how to apply linear algebra; it’s not a course in applications.
Very Good Feature: specialized indices on the endpapers of notation and of examples, in addition to a thorough conventional index in the back of the book.
The book is supplemented with a long list of ancillary materials, mostly for the instructor, including a companion web site, a solution manual for students, Enhanced WebAssign, a test bank, and an instructor’s guide. I did not examine any of these materials.
So: granted that the book is somewhat bland, what’s not to like about it? Certainly the price, at $215 list, is a concern. I might be willing to pay this much for a specialized research monograph that would only sell a few copies and has to make back its fixed costs with a high single-copy price, but it’s ridiculous that a mass-market textbook like this would be so expensive. A good alternative is Strang’s Introduction to Linear Algebra, at less than half the price. It covers the same broad topics, but with more depth and less breadth, and is more application-oriented.
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 MathNerds.org, a math help site that fosters inquiry learning.
Introduction: The Racetrack Game. The Geometry and Algebra of Vectors. Length and Angle: The Dot Product. Exploration: Vectors and Geometry. Lines and Planes. Exploration: The Cross Product. Applications: Force Vectors; Code Vectors. Vignette: The Codabar System.
2. SYSTEMS OF LINEAR EQUATIONS.
Introduction: Triviality. Introduction to Systems of Linear Equations. Direct Methods for Solving Linear Systems. Exploration: Lies My Computer Told Me. Exploration: Partial Pivoting. Exploration: Counting Operations: An Introduction to the Analysis of Algorithms. Spanning Sets and Linear Independence. Applications: Allocation of Resources; Balancing Chemical Equations; Network Analysis; Electrical Networks; Linear Economic Models; Finite Linear Games. Vignette: The Global Positioning System. Iterative Methods for Solving Linear Systems.
Introduction: Matrices in Action. Matrix Operations. Matrix Algebra. The Inverse of a Matrix. The LU Factorization. Subspaces, Basis, Dimension, and Rank. Introduction to Linear Transformations. Vignette: Robotics. Applications: Markov Chains; Linear Economic Models; Population Growth; Graphs and Digraphs; Error-Correcting Codes.
4. EIGENVALUES AND EIGENVECTORS.
Introduction: A Dynamical System on Graphs. Introduction to Eigenvalues and Eigenvectors. Determinants. Vignette: Lewis Carroll's Condensation Method. Exploration: Geometric Applications of Determinants. Eigenvalues and Eigenvectors of n x n Matrices. Similarity and Diagonalization. Iterative Methods for Computing Eigenvalues. Applications and the Perron-Frobenius Theorem: Markov Chains; Population Growth; The Perron-Frobenius Theorem; Linear Recurrence Relations; Systems of Linear Differential Equations; Discrete Linear Dynamical Systems.
Vignette: Ranking Sports Teams and Searching the Internet.
Introduction: Shadows on a Wall. Orthogonality in Rn. Orthogonal Complements and Orthogonal Projections. The Gram-Schmidt Process and the QR Factorization. Exploration: The Modified QR Factorization. Exploration: Approximating Eigenvalues with the QR Algorithm. Orthogonal Diagonalization of Symmetric Matrices. Applications: Dual Codes; Quadratic Forms; Graphing Quadratic Equations.
6. VECTOR SPACES.
Introduction: Fibonacci in (Vector) Space. Vector Spaces and Subspaces. Linear Independence, Basis, and Dimension. Exploration: Magic Squares. Change of Basis. Linear Transformations. The Kernel and Range of a Linear Transformation. The Matrix of a Linear Transformation. Exploration: Tilings, Lattices and the Crystallographic Restriction. Applications: Homogeneous Linear Differential Equations; Linear Codes.
7. DISTANCE AND APPROXIMATION.
Introduction: Taxicab Geometry. Inner Product Spaces. Exploration: Vectors and Matrices with Complex Entries. Exploration: Geometric Inequalities and Optimization Problems. Norms and Distance Functions. Least Squares Approximation. The Singular Value Decomposition. Vignette: Digital Image Compression. Applications: Approximation of Functions; Error-Correcting Codes.
Appendix A: Mathematical Notation and Methods of Proof.
Appendix B: Mathematical Induction.
Appendix C: Complex Numbers.
Appendix D: Polynomials.