# Review of R. Beezer’s “A first course in linear algebra”

Cover of Beezer’s linear algebra book

Speaking of material, what does this book actually cover? The chapters are:

• [SLE]
Systems of linear equations

• [V]
Vectors

• [M]
Matrices

• [VS]
Vector spaces

• [D]
Determinants

• [E]
Eigenvalues

• [LT]
Linear transformations

• [R]
Representations

There are also several appendices (for example, one on Sage a free and open source mathematics software program with very strong linear algebra computational capabilities [S]).

Note the unusual labeling of the chapters – no numbering is used for the chapters, instead capital Roman letters are used (an acronym, E for Eigenvalues, for example). The same indexing method is used for the definitions, examples and theorems as well. How do you refer to a result with this method of labeling? The way cross-referencing works in this book is that each reference is followed by the page number that the reference can be found. For example the theorem that says the adjoint of a matrix respects matrix addition would be referred to as “Theorem AMA [175]” since it can be found on page 175 and is written there as

Suppose $A$ and $B$ are matrices of the same size. Then $(A+B)^T = A^T + B^T$.

One advantage to this method of indexing is that if, for example, replace the chapter on Vectors with your own chapter then recompile the LaTeX, the theorem on adjoint and matrix addition is still “Theorem AMA,” even though its page number has probably changed. Although this indexing method with the page numbers does make the various results pretty easy to find, all the labeled results are also conveniently listed in the index. Moreover, in the the electronic versions (PDF, XML, jsMath) of the book, all of this cross-referencing is available as hyperlinks.

Each chapter of course is composed of many sections. Each section not only has plentiful detailed examples and its own list of “reading questions” to see if you have understood the material, but also a list of exercises with solutions. More precisely, some exercises are marked as computational (e.g., solve the following system of linear equations), some as theoretical (e.g., prove that if a vector v is in the kernel of a matrix then any scalar times v is also in the kernel), and some as “mixed” (often of the form: find an example of a matrix with property X). As far as I could see, every computational exercise was accompanied by a more-or-less complete solution. However, the other types of problems usually (not always) either had no solution provided or only a sketch of one. A rough count would be about 500 exercises and 350 of them have solutions.

The book is very well-written and at a level similar to Anton’s popular book \cite{A}. What topics are actually covered by this book? There is a detailed table of contents of (a) the chapters, sections and subsections, (b) the definitions, (c) the theorems, (d) the examples, and (e) a 30 page index.

The first chapter on systems of linear equations discusses solving systems of linear equations, as well as classifying them (consistent, homogeneous, and so on), row reduction, and the matrix form of a system of linear equations. This chapter, in my opinion, approaches things in the right way by stressing the importance of the method of row reduction/Gauss elimination. Very detailed proofs are also presented. For most students, the more details the better, so this is also a strong plus.

The second chapter on vectors starts with the basics on vector operations. This is followed by four sections on linear spans and linear independence. This is a challenging topic for students but the book does a very though job. The final section discusses orthogonality, inner products and the Gram-Schmidt procedure.

The chapter on matrices comprises six sections. After several sections explaining the usual details on matrix operations (such as transposes and adjoints) and matrix arithmetic (such as addition and multiplication), the discussion turns to matrix inverses. Two sections are spent with this topic. The last two sections deal with row and column spaces and their basic properties.

The chapter on vector spaces begins with two sections on the basic definitions and properties of vector spaces and subspaces. The next two sections are on linear independence (again), spanning sets (again), and vector space bases. With the difficulty that linear independence and spanning sets present to many students, this reappearance of these topics provides excellent reinforcement. The last sections of the chapter are on the dimension properties of the row-span and column span of a matrix. For example, what Gilbert Strang calls the Fundamental Theorem of Linear Algebra, the so-called “rank plus nullity theorem,” is in one of these sections (see Theorem RPNC in section RNM on page 323).

After a relatively short 24-page chapter on the determinant and its properties, the book continues with the Eigenvalues chapter. This chapter discusses eigenvalues, eigenvectors, diagonalization, and similarity. Topics such as the Jordan canonical form are discussed in a later chapter.

The chapter on linear transformations has separate sections on injective maps, surjective maps, and invertible maps, resp.. This chapter has a wealth of examples to help guide the student through these relatively abstract concepts.

The final chapter is also the most difficult for the student. This is the chapter titled Representations, concerning topics such as the matrix representation of a linear transformation and the properties of this representation. Other topics, such as the Cayley-Hamilton Theorem, the Jordan canonical form, and diagonalization of orthogonal matrices appear in this chapter as well. In version 2.0, some sections of this chapter were less complete than those in other chapters.

Finally, I would also like to mention that the author has done a considerable amount of work on the use of Sage in the classroom. For example, he has added to the Sage code and documentation and has written the following “quick reference sheets” summarizing the most useful linear algebra commands in Sage. For more information, see his website http://linear.ups.edu/sage-fcla.html.

An excellent book written by an experienced teacher. Highly recommended.

Bibliography:

[A] H. Anton, Elementary linear algebra, 8th edition, Wiley, 2000.

[B] Robert Beezer’s Linear Algebra website, http://linear.ups.edu/.

[S] W. Stein, Sage: Open Source Mathematical Software (Version 5.2), The Sage~Group, 2012, http://www.sagemath.org