Ward Cheney and David Kincaid have produced a number of excellent textbooks together. Their new linear algebra textbook, Linear Algebra: Theory and Applications, continues this tradition of excellence. True to its subtitle, the book contains all the theoretical content one would expect in an undergraduate linear algebra textbook as well as a rich assortment of applications. As the authors explain in the preface, one could select material from the book to teach a one-semester course emphasizing either theory or applications. Alternatively, one could cover the entire book in two semesters for a multi-faceted course in linear algebra covering algebraic theory, practical applications, mathematical software (i.e. MATLAB, Mathematica, and Maple), and an introduction to numerical linear algebra.
Linear Algebra reads easily. It is written in a conversational style that draws the reader in. You get the feeling that the authors are not in a hurry. They provide a generous amount of expository prose and pause along the way to share witty quotes and give interesting historical footnotes. Even though the book proceeds at a leisurely pace, it covers a great deal of material. It covers the standard topics as well as several others not always included in an undergraduate course such as singular value decomposition, Gerschgorin's theorem, and iterative methods for solving linear equations.
Many books with the subtitle "Theory and Applications" are almost exclusively devoted to theory. This is not the case with Linear Algebra. Cheney and Kincaid's text contains mathematical applications of linear algebra such as partial fraction decomposition, least squares, and interpolation. It also contains numerous applications of linear algebra outside of pure mathematics such as balancing chemical equations, analyzing circuits with Kirchhoff's laws, and explaining the Google PageRank algorithm.
At first, the size of Linear Algebra is surprising. At 740 pages, it is about the size of many calculus textbooks. One could write a terse definition-theorem-proof linear algebra text book in far fewer pages. But Linear Algebra is big for good reasons. It contains many examples and applications as well as thousands of exercises. It also covers aspects of linear algebra often omitted from more succicnt textbooks and is a pleasure to read.
John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs at The Endeavour.
Chapter 1: Systems of Linear Equations
1.1 Solving Systems of Linear Equations
1.2 Vectors and Matrices
1.3 Kernels, Rank, Homogeneous Equations
Chapter 2: Vector Spaces and Transformations
2.1 Euclidean Vector Spaces
2.2 Lines, Planes, and Hyperplanes
2.3 Linear Transformations
2.4 General Vector Spaces
Chapter 3: Matrix Operations
3.2 Matrix Inverses
Chapter 4: Determinants
4.1 Determinants: Introduction
4.2 Determinants: Properties
Chapter 5: Vector Spaces
5.1 Column, Row, and Null Spaces
5.2 Bases and Dimension
5.3 Coördinate Systems
Chapter 6: Eigensystems
6.1 Eigenvalues and Eigenvectors
Chapter 7: Inner Product Vector Spaces
7.1 Inner Product Spaces
Chapter 8: Additional Topics
8.1 Hermitian Matrices and Spectral Thm.
8.2 Matrix Factorizations and Block Matrices
8.3 Iterative Methods for Linear Equations