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Linear Algebra as an Introduction to Abstract Mathematics

Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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Here at Iowa State University, we offer two separate and mutually exclusive introductory courses in linear algebra. One, at the sophomore level, is intended for people who are not mathematics majors; it emphasizes computation and applications. Majors take a junior-level course that focuses on proof. This book is a viable candidate as a text for the latter course, or any other course in which linear algebra is taught from a proof perspective, but the succinctness of the exposition may give an instructor pause.

The book is quite a slim one (less than 200 pages of text, around 70 pages of which are appendices), focusing primarily on material that has now become fairly standard for an introductory course, although, as we’ll soon see, there are a few things found here that are not likely to be covered in many first courses. The text starts with an introduction (“What is Linear Algebra?”) that describes the subject by focusing on the solution of linear equations. The next chapter is a fairly complete introduction to complex numbers (defined as ordered pairs of real numbers) and their arithmetic. Somewhat surprisingly, the book even offers, in chapter 3, a proof of the Fundamental Theorem of Algebra, assuming as known a preliminary result on attainment of extreme values on the closed unit disc. This, of course, is one of the “things found here that are not likely to be covered in many first courses” that I referred to at the beginning of this paragraph. I’m not sure that a first course in linear algebra is the time and place for a proof of the Fundamental Theorem, and if teaching our proof course from this book, I would almost certainly skip this; however, I am also loath to criticize a book for including too much interesting mathematics.

After this preliminary material, the book gets down to linear algebra. There is a chapter on vector spaces (over the fields of real or complex numbers; arbitrary fields are not used), followed by one on linear independence and spanning sets, together with a proof that any finite-dimensional vector space has a basis. This is then followed by two chapters on linear transformations and eigenvalues/eigenvectors. The chapter on linear transformations discusses the matrix corresponding to a linear transformation relative to a given basis, but the issue of changing bases is deferred until later. Matrix arithmetic and its connections with systems of linear transformations, by the way, are not discussed in the body of the text itself, but in an appendix.

Eigenvalues and eigenvectors are defined for transformations, with connections to matrices discussed. In particular, it is proved that any linear operator on a complex finite-dimensional vector space can be put into upper triangular form. Because determinants have not yet been introduced, the characteristic polynomial is not, at this point, defined, but an early preview of the concept is given for 2 x 2 matrices.

The authors prove the existence of eigenvalues for operators on complex vector spaces by a determinant-free argument that is used in Axler’s Linear Algebra Done Right, but the book does not seem to be as stridently anti-determinant as is Axler’s text; determinants are not deferred until the very end of the book, as is the case with Linear Algebra Done Right, but are instead discussed in the very next chapter. It is in this chapter that the characteristic polynomial is defined; it is mentioned that the roots of this polynomial are the eigenvalues of the matrix, but no specific computations are given.

Inner product spaces are discussed in the next chapter. All the usual material is covered here, including orthogonal projection and (quite briefly) its application to minimization problems. However, the phrase “method of least squares” is not used.

Next, there is a very brief chapter on how the matrix of a linear transformation changes with regard to a change in basis. Then, in chapter 11, the text returns to inner product spaces, discussing operators on them. Self-adjoint and normal operators are defined, and a version of the spectral theorem for normal operators (that an operator is normal if and only if it is represented by a diagonal matrix with respect to an orthonormal basis) is established. The chapter ends with a very brief look at positive operators, the polar decomposition, and the singular value decomposition. This entire chapter is only about ten pages long, so clearly none of these ideas can be developed in depth, but that can hardly be expected in a text for a first course in linear algebra, most of which don’t discuss this material at all.

There are four appendices. The first is a set of supplementary notes on matrices and their applications to linear systems; the second discusses the basic terminology of sets and functions; the third introduces some basic algebraic terminology; the fourth lists some common mathematical symbolism and, in an unusual but very much appreciated feature, provides historical commentary as well. Readers who want to know, for example, the origins of the equal sign, or the symbol \(\infty\) for infinity, will find answers here.

Although the approach taken in this book is proof-oriented, this is not a text on advanced linear algebra. As the above summary of the book’s contents should indicate, the topics covered here are typically those of an introductory course, and standard second-semester topics such as the minimal polynomial, canonical forms, Cayley-Hamilton theorem, etc. are not discussed.

The book is written so that it should be understandable by anybody who has taken a semester or two of calculus (and even that prerequisite is not strictly necessary, since the only time calculus is actually used is in some examples). However, the exposition is fairly succinct, and some prior exposure to proofs would likely be very valuable. The book states in the preface that that elusive and ill-defined concept, “mathematical maturity”, is the real prerequisite for the book.

And that observation leads me to what I think is the one real difficulty I have with the text. Since “mathematical maturity” means different things to different people, the book would have benefited, I think, from some discussion of basic logic and proof techniques, if only to explain to the students, for example, how to negate a simple mathematical sentence, and what the converse and contrapositive of a statement is. This seems particularly appropriate for any book that refers to itself as an “introduction to abstract mathematics”, and, given that the book is certainly not unduly long, a 15 or 20 page discussion along these lines could easily have been accommodated, perhaps as another appendix.

Other mild quibbles: there is no bibliography at all in the text, and it probably wouldn’t have hurt to spend more time with detailed discussion of examples. In particular, I would like to have seen examples of infinite-dimensional spaces, combined with some indication of how one proves that a vector space is not finite-dimensional.

It should be noted that, at least as of this writing, a free PDF copy of the book can be obtained at one of the authors’ webpages. The page numbers between the PDF and the printed text don’t match up exactly, there are more appendices in the PDF copy, and a couple of figures that are shaded in the text are in color in the PDF. Other than these very minor differences, the PDF appears nearly identical to the hard copy.

To summarize and conclude: this is a well-written book, but the succinctness of exposition and failure to discuss proof techniques would likely lead me to find a different text for Iowa State’s proof-based linear algebra course. However, this book might fare better as a potential text for an honors course in linear algebra.

Mark Hunacek ( teaches mathematics at Iowa State University. 

  • What is Linear Algebra
  • Introduction to Complex Numbers
  • The Fundamental Theorem of Algebra and Factoring Polynomials
  • Vector Spaces
  • Span and Bases
  • Linear Maps
  • Eigenvalues and Eigenvectors
  • Permutations and the Determinant of a Square Matrix
  • Inner Product Spaces
  • Change of Bases
  • The Spectral Theorem for Normal Linear Maps
  • Appendices:
    • Supplementary Notes on Matrices and Linear Systems
    • The Language of Sets and Functions
    • Summary of Algebraic Structures Encountered
    • Some Common Math Symbols and Abbreviations