This is a too-long but otherwise reasonable introduction to linear algebra. The book’s great strength is that it eases very gradually into the subject, starting with very simple, concrete examples of vectors and matrices in **R**^{2} and **R**^{3} and gradually working into abstract vector spaces, with lots and lots of numerical examples and exercises throughout. The book’s great weakness is that it treats these subjects in almost complete isolation from the rest of mathematics and from applications. The book takes a “taxonomy” approach: it defines some objects and some properties, categorizes the objects according to which properties they have, and then declares the job done. There’s no indication of why these properties are useful or interesting.

The book’s length comes primarily from developing everything three times: first a very detailed study of **R**^{2} and **R**^{3}, then a re-development of these same properties in a mixture of abstract vector spaces and **R**^{n} (slanted heavily to the latter and to finite-dimensional vector spaces, but with some infinite-dimensional examples), and finally a somewhat-abbreviated development of vector spaces over **C**.

A couple of other features help the book bulk up. The “Key Concepts” at the end of each section unfortunately do not single out the concepts that are key, but list every definition and result in the section; they are essentially a repeat of the section, but without the discussion and the examples. The number of drill exercises is probably excessive; most of the exercises have answers in the back of the book.

There are a couple of conspicuous omissions. The *LU*-decomposition is not mentioned, although *QR*-decomposition and Cholesky decomposition are covered thoroughly. There is no discussion of calculators, computers, or technology, except in the Singular Value Decomposition section. The book assumes you will do everything by hand, and even in the SVD section it only says “use technology” without giving any details.

The book also has a few quirks. It is published as a 3-ring binder, which is peculiar but might be useful, because it allows you to separate out the pages of interest at the moment and not carry around this heavy book. Venn diagrams are drawn as colored solid parallelograms instead of circles. There is a long Chapter 0 that deals with logic and set theory. Most of this is conventional, except that the definition of “logical statement” requires that we be able to determine whether the statement is true or false; the Goldbach Conjecture, for example (and all unsolved math problems) is not a logical statement. It seems impossible to build a consistent logic on such a definition, because whether a statement is logical or not depends on what we know, not on the statement itself, and this changes all the time. Happily the logical statement concept seems not to be used outside of Chapter 0.

Bottom line: Not a bad book, but quirky and probably overkill for most audiences. A more concise, thorough, and conventional book such as Strang’s Introduction to Linear Algebra would be better for most uses.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

## Comments

## Mary McLellan

As a student, what I like most about

A Portrait of Linear Algebrais the order in which topics are presented, and the extensive practice with proofs in the exercises. The topics revisit and build on the early foundational chapters so that new concepts develop naturally throughout a course. The proof exercises require critical thinking about connections within Linear Algebra and beyond.## Patricia Michel

In

A Portrait of Linear Algebra, author Jude Thaddeus Socrates takes us on a lively tour through the world of introductory linear algebra. In this well-written and entertaining text, Socrates - a Pasadena City College professor who holds a Ph.D from Caltech - relates his appreciation for the beauty of mathematics in an approach that is refreshing and well-conceived.The book begins with a Zero Chapter which reintroduces the student to the basic principles involved in a mathematical proof. Having set up the foundation for the rigor that is to follow, Socrates proceeds to review and make abstract, topics that the student has already experienced - namely Euclidean spaces. Chapters 1 and 2 are devoted to ideas and definitions related to these familiar spaces. The diagrams in these two chapters and throughout the book are noteworthy in their effectiveness at clarifying the concepts being discussed. Examples and homework problems in the text are interesting and creative while ranging in difficulty from basic to demanding.

After laying the groundwork of what properties give structure to R

^{n}, Socrates delves into abstract vector spaces and inner product spaces with all their peculiarities. The discussion throughout the text is rigorous and animated, always referring back to what has already been learned. The examples build on themselves and increase in complexity. As topics become more and more detailed, a larger and clearer picture of linear algebra with all its nuances is developed. For the student interested in continuing in mathematics this book provides a wonderful bridge to the world of advanced mathematics.