How can one make first-semester linear algebra material more appealing (or at least more palatable) to leery students? Why not try dressing it up in a story of young love? For those unfamiliar with the term, *manga* refers to a type of Japanese comics. *The Manga Guide to Linear Algebra* is just one in a series of manga presentations of scientific material such as statistics, molecular biology, and, perhaps most ambitiously, the universe. This particular guide seems aimed at high school and college students, though it could also appeal to general readers looking for a gentle refresher course on the topic.

The book’s main characters are mathematically-talented college student Reiji, his fetching tutee Misa, and Tetsuo, Misa’s older brother. The storyline is simple: Tetsuo, the head of a karate club, allows Reiji to join the club on the condition that Reiji tutors Misa in linear algebra. The book details the tutoring sessions, while providing occasional interludes which either further the book’s story or provide supplemental mathematical material.

The literary quality of the work is about what you’d expect of a book that is primarily a mathematical text: the plot is superficial and predictable, though I enjoyed the author’s successfully drawn parallel between Misa’s efforts to learn linear algebra and Reiji’s efforts to learn karate. I was also mildly disappointed in the book’s stereotyped gender roles: while Misa is intelligent, she experiences no personal growth in the book (in contrast to Reiji, and even Tetsuo). As a woman in the field of mathematics, I would have preferred it if Misa’s main talents hadn’t been making lunch for Reiji and looking pretty.

The book’s mathematical discussions range from elegant to confusing. Its author clarifies that the book is intended as a “complement to more comprehensive literature, not a substitute,” and as such it is moderately successful. Many introductory linear algebra topics, as well as preliminary material, are reviewed in the book, and the author takes advantage of the manga format to provide some deft visual demonstrations of techniques and concepts. I also like that the book discusses some more abstract areas of linear algebra, such as linear independence and subspaces of **R**^{n}.

That being said, the book should certainly not be used as a solo text by someone who hasn’t already studied the material. Many algorithms or concepts that students find difficult are rushed through. For instance, when Reiji covers Gaussian elimination, he provides only two examples, and both involve matrices whose non-augmented versions reduce to the identity matrix. The book’s scope is limited — it focuses primarily on square, invertible matrices, and contains little to no discussion of inconsistent systems of linear equations or of those systems that have more than one solution. The book’s method of computing determinants for general square matrices seems unnecessarily convoluted, requiring the explicit use of permutations.

At least once the text states something that is patently false: Reiji tells Misa that you “can generally never find more than *n* different eigenvalues and eigenvectors for any *n *×*n* matrix,” which is true about eigenvalues but false about eigenvectors: every eigenspace of every matrix contains infinitely many elements. While this may be merely the result of a mistranslation, this is the type of error that can seriously confuse someone who isn’t already comfortable with linear algebra.

Thus, while I applaud the unique format of the book, and while portions of it provide solid summaries of linear algebra techniques, the book is less than stellar in its execution. It could work well as a study guide for someone who just needs a review of linear algebraic concepts, but I hesitate to recommend it to someone, like Misa, who is struggling with the subject. That said, I encourage authors and publishers to continue publishing creative texts such as this.

Jessica K. Sklar is an associate professor of Mathematics at Pacific Lutheran University. Trained as an algebraist, she considers herself more of a generalist these days, with particular interest in recreational mathematics, and in the communication and writing of math. She recently coedited (with Elizabeth S. Sklar) the collection of essays *Mathematics in Popular Culture**: **Essays on Appearances in Film, Fiction, Games, Television and Other Media** *(McFarland & Co., 2012), and in 2011, she and coauthor Gene Abrams (University of Colorado at Colorado Springs) received an MAA Carl B. Allendoerfer Award for their article “The Graph Menagerie: Abstract Algebra and the Mad Veterinarian” (*Mathematics Magazine*, June 2010). Her homepage is http://www.plu.edu/~sklarjk and you may contact her at sklarjk@plu.edu.

## Comments

## Donald L. Vestal

Karate, Math, and Romance! In this Manga Guide, Reiji seeks to learn karate. The captain of the karate team, Tetsuo, agrees to let Reiji join if he tutors his sister, Misa, in linear algebra, which is where the math comes in. As it turns out, Reiji has a crush on Misa, and that's where the romance comes in.

Linear algebra is presented here by way of the linear transformation. Topics covered are: matrices, vectors, linear independence and spanning, subspaces and dimension, linear transformations, and eigenvalues and eigenvectors. The text is heavy on computation and light on abstract theory. Most of the examples are done in two and three dimensions, allowing pictures to be used to show the geometry of linear transformations. The only drawback is that the book is light on applications. But if you're a visual learner and you want to learn linear algebra, this just might be the book for you!