The word “advanced” in the title of this book can be interpreted in two ways. On the one hand, the book discusses topics in linear algebra (canonical forms, bilinear and sesquilinear forms, matrix Lie groups, etc.) that are rarely taught in an introductory course. On the other hand, even when discussing topics that are usually considered elementary, the book does so in a theoretical, sophisticated way, generally eschewing the kind of routine calculations that are so common in introductory linear algebra books in favor of carefully defined terms and precise statements (and proofs) of theorems, often presented at a fairly high level of generality.
This approach is evident even in the very first chapter, on vector spaces and linear transformations. Vector spaces are not assumed finite-dimensional, and are defined over an arbitrary field rather than the field of real or complex numbers. The term “field” is not defined in the text, thereby making it clear even from page 1 that the author assumes that the reader has some background in abstract algebra. Zorn’s Lemma is also assumed, and used to prove the existence of a basis for an arbitrary vector space V. Likewise, the author’s discussion of the fact that two bases for V have the same number of elements also does not assume finite-dimensionality (though things are simplified somewhat by not distinguishing between cardinalities of infinite sets). This chapter also discusses dual (and double dual) spaces and the dual T* of a linear transformation T.
After a chapter addressing coordinates and the matrix of a linear transformation, (including discussion of change of basis and the matrix of the dual transformation), the book moves to determinants, which are initially given a somewhat nonstandard axiomatic geometric definition: axioms are given for the concept of a “volume function” which associates to every n x n matrix a real number (representing the volume of the “parallelogram” spanned by the columns of the matrix). It is shown that a volume function exists and is unique up to scaling, and the determinant of a square matrix A is defined to be the image of A under that volume function (scaled, of course, so that the volume of the unit n-cube is 1). This definition is immediately followed by two sections, giving a more traditional characterization of determinants and setting out the basic facts about them. The chapter then concludes with three sections discussing topics not generally seen in elementary courses: the first discusses determinants and invertibility of integer matrices, the second discusses orientation in real vector spaces (via a detour through the topology of the general linear group in which the connected components of the group GLn(R) are characterized; knowledge of the topological ideas is assumed), and the third discusses (without proof) some basic facts about Hilbert matrices.
This approach to determinants reflects a fairly common theme: the author frequently gives nonstandard definitions of topics and then proves the equivalence to the usual definition. For example, to define the product of two matrices A and B (of appropriate sizes) the author takes the composition of the linear transformations defined by A and B and defines AB to be the matrix corresponding to this composition. Likewise, the transpose of a matrix A is defined as the matrix corresponding to the dual of the transformation induced by A. Whether any enhanced motivation and insight given by this approach justifies the departure from standard definitions is, of course, a matter of individual taste.
The next two chapters address the structure of linear transformations via eigenvalues and canonical forms. The first of these chapters provides a very succinct and efficient development of eigenvectors and generalized eigenvectors (developed in tandem), the minimal and characteristic polynomials of a matrix, diagonalizability and triangularizability, and then culminates in a section relating these concepts to the theory of linear differential equations. The next chapter (the longest and, I thought, most difficult) addresses the Jordan and rational canonical forms of a matrix. (The approach here is by invariant subspaces; in an appendix, the author discusses the module-over-a-PID approach to this topic.) An algorithm for computing the Jordan form of a matrix (whose characteristic polynomial is given as the product of linear factors) is provided, as are several very helpful, rather non-trivial, examples, worked out in some detail. It would have been nice, though, to see more applications of the material in this chapter; for example, it would not have taken much additional exposition to prove the interesting result that any square matrix is similar to its transpose, but this does not appear.
Chapters 6 and 7 discuss, respectively, forms (bilinear, sesquilinear, quadratic) and their classification, and inner products on real and complex vector spaces. Unlike many books on quadratic forms (e.g., Lam’s Introduction to Quadratic Forms Over Fields) the author here does not adopt the blanket assumption that char F ≠ 2. The discussion of inner products in chapter 7 addresses many familiar topics (Gram-Schmidt, normal and unitary operators, spectral theorem), done clearly and succinctly, with attention paid to the infinite-dimensional situation and nice applications to analysis and topology given. A final section discusses singular values; consistent with the author’s strong belief that linear algebra “is about vector spaces and linear transformations, not matrices” (about which, more later) the standard decomposition A = U∑VT does not, however, appear; everything is phrased in terms of eigenvalues of transformations.
The final chapter is entitled “Matrix Groups as Lie Groups”. Recent years have seen the publication of a number of books (Stillwell’s Naive Lie Theory being an excellent example) that attempt to make the rudiments of Lie theory accessible to a broader audience by working with matrix groups rather than general Lie groups, thereby exposing the reader to the ideas of Lie theory in a concrete (but reasonably general) setting, letting linear-algebraic arguments replace more difficult manifold-theoretic ones. Given that this is a book on linear algebra, I expected, when I saw the title of this chapter, that it was intended to do the same. However, the chapter begins with the definition of a Lie group (the author assuming knowledge of the definition of differentiable manifold and other facts about differential topology) and consists primarily of the definitions of various classical matrix groups and proof that they are Lie groups. Aside from the fact that many people who have studied differentiable manifolds probably have already seen most or all of these examples, this chapter seems like a lost opportunity to showcase the utility of linear algebra in learning something about Lie theory, perhaps by talking about the exponential of a matrix and the Lie algebra corresponding to a matrix group.
I enjoyed reading this book and more than once found myself admiring the exposition, but I am not sure I know just who its intended audience is. Since there are no exercises at all, it seems not to be intended as a text; the fact that a number of terms are given non-standard definitions may also limit its value in this regard. The book’s value as a reference may be limited by the choice of topics.
As noted earlier, the author firmly believes that linear algebra is not about matrices; he says so both in the preface and the body of the text. People are of course entitled to their opinion as to whether linear algebra is best approached from the operator-theoretic or matrix-theoretic viewpoint, but it seems a bit odd to baldly assert as fact a statement like this, particularly given that (a) much of the area of numerical linear algebra is concerned with matrices, (b) many people do research in matrix theory and consider themselves linear algebraists, and (c) journals like Linear Algebra and its Applications run lots of articles about matrices. In any event, a number of matrix-oriented topics (Perron-Frobenius, Rayleigh-Ritz, Courant-Fischer, Gershgorin) that are covered in some graduate linear algebra courses are not mentioned in this book at all, thereby making this book less attractive as a potential resource for graduate students preparing for qualifying examinations. However, students preparing for exams that do not cover these topics, or mathematicians not specializing in linear algebra who want a concise reference for the topics that are covered, might well find this book worth a serious look.
Mark Hunacek teaches mathematics at Iowa State University. After near-simultaneous acquisitions of both a PhD and a wife, he solved the “two body problem” in his family by going to law school and then becoming an Assistant Attorney General for the state of Iowa while his wife pursued a career as a mathematics professor. He is happy to report, however, that he has now retired from the practice of law and returned to the fold of mathematics teaching (but he also teaches a course in engineering law for old time’s sake.)
1. Vector spaces and linear transformations
4. The structure of linear transformations I
5. The structure of linear transformations II
6. Bilinear sesquilinear, and quadratic forms
7. Real and complex product spaces
8. Matrix groups as Lie groups
B. Modules over principal ideal domains