As indicated in the latter part of his title, Edgar Goodaire takes a “pure and applied” approach to learning Linear Algebra. Along with rigorous presentations of theorems and proofs, there are some more applied topics such as the pseudoinverse of a matrix in chapter 6 and the Singular Value Decomposition in chapter 7. Goodaire stresses he wants nothing in the book to be stated without justification. To help students with the proofs, he has included an appendix called “Show and Prove” and “Things I Must Remember” which were complied by his students over the years of writing the book.

Goodaire begins the book by presenting vectors in \(\mathbb{R}^2\) and \(\mathbb{R}^3\). He does this because he wants to convey to the reader that there is a difference between Linear Algebra and Matrix Algebra. The latter, he says, includes material on matrix multiplication and solving systems of equations. Linear Algebra is much more, however, and he doesn’t want readers to think that Linear Algebra is purely a computational subject. Section 1.1 starts with the fundamental properties of vectors such as addition, scalar multiplication, and graphing vectors. There is also an introduction to some of the major concepts in Linear Algebra such as linear combinations, span, and bases. While these introductions are good, some professors may prefer to see them in a later chapter. The same intermingling occurs in section 1.2. The section starts with the length or magnitude of vectors, but then dives into talking about standard basis vectors.

If there is enough time in the semester to get to orthogonality and the Spectral Theorem, then the final two chapters of the book are great for further and deeper study of Linear Algebra. However, they can also be used for independent study.

In the Linear Algebra books I have read, I see two approaches to presenting material. One approach is to have topics laid out in order with no bouncing back and forth. The other introduces some of the more theoretical ideas within the first chapters and then coming back to these topics in the middle of the text. (These topics are linear independence, span, vector subspaces, row- column- and null-spaces, and linear transformations.) This book is an example of the bouncing back and forth approach, but in a way that is not as overwhelming as, for example, in David Lay’s book. Some professors may prefer not to see this preview of topics embedded in early chapters; this is the approach in the books by Holt, Fraleigh/Beauregard, and Larson.

Goodaire’s book gives a proper balance to logic and computation. This book is worth considering if you feel the need to change books for your course.

Peter Olszewski is a Mathematics Lecturer at Penn State Erie, The Behrend College. His research fields are in mathematics education, Cayley color graphs, Markov chains, and mathematical textbooks. He can be reached at pto2@psu.edu. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes.