Review of Gilbert Strang's Linear Algebra and Its Applications, 3rd edition, 1988.

Reviewed by Tom Hagedorn

Strang's textbook has been widely used for many years (the author himself learned linear algebra from this text) and not surprisingly, its organization is fairly traditional. There are eight chapters in the book, but a standard first course in linear algebra will probably cover only the first five. The topics of these chapters are, in order: Matrices and Gaussian Elimination, Vector Spaces and Linear Equations, Orthogonality, Determinants, and Eigenvalues and Eigenvectors. As the context for most of Strang's text is Rⁿ, and he spends time explaining its geometry, the placement of orthogonality before that of eigenvalues is quite natural. Overall, Strang's text contains many applications from science and engineering and his text seems well-suited for an audience with a good scientific knowledge. Many of the worked examples, some of which take several pages, require a good familiarity with science. Students, and instructors, without such knowledge may have difficulty with the text. Also, though most results are justified by an informal discussion in the text, students may have a hard time determining what is and is not proven, and so this text does not seem suitable for an course which is intended to introduce students to proofs.

It is interesting to study Strang's development of linear algebra in his first two chapters. Strang's textbook begins by a thorough discussion of solving linear equations and the geometry of their solutions. Matrices and their operations are then introduced. Elimination theory then leads to a discussion of the LDU decomposition followed by a discussion of the inverse of a matrix and how to compute them via the Gauss-Jordan method and how to determine which matrices are invertible via the use of non-zero pivots. Transposes are also introduced and discussed. The last section of Chapter 1 introduces an application of using linear algebra to find an approximate solution to a differential equation. Though I had fond memories of this text, in comparison with some of the other texts on the market (e.g. Lay) this part of the text seemed awkward. While the introduction of matrices is well-motivated, the introduction of matrix multiplication in the context of row operations, in the absence of any discussion of linear transformations, seems unnatural. Similarly, the presentation of inverses and transposes seems perfunctory and unrelated to the other ideas of the chapter. Other texts seem better able to present these concepts in a unified manner.

The real strength of Strang's text, I had often thought, was in Strang's development of vector spaces in Chapter 2. Strang immediately introduces the definition of abstract vector spaces and presents many examples in addition to the space of n-dimensional vectors. He then discusses the solution of m equations in n unknowns in terms of the dimension of subspaces and the rank of matrices. He then develops the concepts of linear independence, basis, and dimension and discusses the four fundamental subspaces in the context of Rn. After a section on an application of linear algebra to graphs and networks, in the final section linear transformations are discussed. For all these concepts, they are discussed in the context of Rn and the examples in the context of other vector spaces are not usually given. Again, as for the first chapter, I had misgivings about the exposition. For example, I have always found Strang's discussion of the four fundamental spaces very wordy and ultimately confusing to students. And the application of linear algebra to graphs and networks, while interesting to someone who knows linear algebra, seems to unnecessarily break up the unity of the chapter.

Overall, Strang's Linear Algebra and Its Applications contains a wealth of applications and examples of how to use linear algebra in science and engineering. I would heartily recommend it as a reference book for anyone who has already taken a course in linear algebra, but I would have trouble recommending it as a text for a first course in linear algebra. Beginning students will be confused by it and while more advanced students may be able to see the big picture, linear algebra can and should be presented as a more unified subject than it is here.