"Visual Linear Algebra is a new kind of textbook," the authors tell us in the Preface. Actually, the book looks very much like any other linear algebra text, until you notice that inside the front cover is hidden a CD. Reading the Preface, one learns about the tutorials provided in this CD in the format of Maple worksheets and Mathematica notebooks; these are to be the center of the student's learning process. The significance of the CD and the justification of the above comment of the authors in fact become much clearer when one takes a look at the Table of Contents: Out of the forty-seven sections of the text, the thirty that are marked with a blue semidisk are exact text versions of the tutorials provided in the CD. Thus, the computer component of a course using this textbook will not be a mere supplement; rather, it is going to be the core of the whole experience. In fact, the pages corresponding to the tutorial sections have blue margins and the book looks mostly blue from the side! The idea is that the examples that use the computer will be more interesting since they can be selected to be computationally more challenging and hence involve more complex systems.
Linear algebra is undoubtedly very basic and important in the undergraduate mathematics curriculum, and often it follows calculus in the list of courses to be taken by students who will not major in mathematics but will need to have a good mathematical foundation in their own majors. Especially for such students, but definitely also for math majors, a good linear algebra course needs to emphasize numerous examples and has to find an approach that will work for a wide variety of learning styles; a purely axiomatic development will not be sufficient. As one way to deal with this situation, various texts and instructors find it useful to spend a considerable time on vectors in two and three dimensional Euclidean spaces in order to help students visualize various concepts of linear algebra; the axiomatics may or may not follow eventually depending on the needs of the students. Visual Linear Algebra also makes extensive use of two and three dimensional examples. The axiomatics part is not postponed until the last few chapters, however; the concepts developed in any tutorial by the help of the visual tools of two and three dimensions are generalized and given a rigorous foundation always within the same chapter.
The text is intended for a one-semester linear algebra course, and the usual mathematical maturity of a student taking a standard first course in linear algebra will suffice to follow it. The subjects covered are the usual bunch; here is the list of the titles of the eight chapters: Systems of Linear Equations; Vectors; Matrix Algebra; Linear Transformations; Vector Spaces; Determinants; Eigenvalues and Eigenvectors; Orthogonality. All the standard concepts of linear algebra are covered, albeit in a somewhat abstract style, in purely textual sections. The use of the computer algebra systems in the tutorial sections allows the authors to add on colorful examples to the text involving Markov processes and more general discrete dynamical systems. There are several sections included which explore interesting applications of linear algebra like computer graphics, cryptology, loops and spanning trees. I was especially intrigued and excited by the use of movies, animated pictures showing how a particular system evolves as some parameter is changed. There is even an exercise where the student is taken through a step-by-step solution of the problem of creating a realistic animation of a car moving along a curved road, taking into consideration the fact that the car needs to be pointing in the right direction!
Incorporating Maple and Mathematica into the text may have one additional benefit; students who use this text will end up getting quite comfortable with the system that they decide to use. This is clearly not a major goal of the text but may be seen as a pleasant by-product. Clearly not all students will come to their first linear algebra course proficient in either of the two computer algebra systems. However, these are both well-documented and widely available computer algebra systems, and it will not take great expertise in either to be able to use and benefit from the tutorials. The time the student uses to learn the basics of either of Maple and Mathematica will be time well-spent; these programs have wide applications and students will most likely come across their various uses further along their careers.
The philosophy underlying this whole project is emphasized in the Preface: the student will only learn by doing. The student is expected to go through a wide variety of exercises; the tutorials allow modifications as well, and thus encourage further exploration of the topics. This indeed is a new kind of textbook, and it tackles with the difficult task of naturally incorporating computers into the standard linear algebra course in order to enhance student participation and understanding. This reviewer believes that this job has been done very well. The few typos in the tutorials and the somewhat unlucky choice of color scheme (blue, gray and black) for the various tips of vector arrows used in the illustrations throughout the text can hopefully be dealt with in a new edition.
Gizem Karaali teaches at the University of California in Santa Barbara.