Customer Reviews

Practical Linear Algebra: A Geometry Toolbox

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I have to say this book was perfect. It's written with the assumption that you know algebra and trigonometry (you'll also need a little bit of calculus if you want to completely absorb chapter 18) and goes from there to describe linear algebra step-by-step. I was struggling with concepts like Eigenvectors, Gaussian elimination, matrix inversion, etc. but after reading through this book and working all the exercises (most of which have answers in the back for self-study), I'm actually finding myself "thinking" in linear algebra.

The first third or so of the book covers 2D linear algebra, and has a bit of a bias towards graphics problems. It covers things like line intersections and "closest point to a line" as well as rotations, shears, translations, etc. The next third or so extends these concepts out to 3D (still with sort of a graphics bias) and introduces 3D-only concepts such as the cross product. Finally, the last third introduces the abstract N-dimensional perspective that doesn't have a graphical interpretation. This is where it discusses things like least-squares estimation and orthonormalization - the really useful (but abstract) bits of linear algebra.

With the first two thirds of the book to back it up, I found the really abstract concepts (which most authors seem to want to start with) relatively easy to absorb - which is really a pretty amazing accomplishment.

I also can't recommend the chapter exercises highly enough. Most of them had answers in the back, so you can check your work (including these ought to be a no-brainer for writers of math books, but evidently there are quite a few math book writers with, well, no brains...) There was a perfect mix of "check your understanding" type questions and "stretch your brain a bit" exercises, but the book itself was entirely self-contained; as long as you're comfortable with basic trigonometry, you'll have no trouble figuring out the answers to the authors questions with just the material in the chapter.

This is a very good book for learning Linear Algebra. It succeeds is two important ways. One it helps you build an intuitive understanding of the subject, something more theoretical books sometimes fail to do. Second, it helps you build a nice toolkit for working with 2D and 3D geometric models.

So, it covers the normal topics of Linear Algebra, vectors, matrices, determinants, Gaussian elimination, eigenvalues, etc. But, it also mixes in topics from Computer Graphics such as points, lines, planes, polygons and Bezier curves. By doing so, it demonstrates just how useful Linear Algebra is for geometric transformations such as scaling, rotation, and translation.

Math majors will want to go beyond this book. There are several books that give a more comprehensive treatment of this subject. But, even to those readers, I suggest you try this one first.

Linear algebra has traditionally been the class in the undergraduate math curriculum where the student makes the transition from "plug - n - chug" formula popping to mathematical proofs. However, in recent years linear algebra has taken on a new role, namely as the best class where applied mathematics can be visually demonstrated. The enormous advances in animated imagery have led to movies where the characters are a virtual hybrid of animated and real. I once taught a course in computer graphics for computer programmers and they were impressed when they applied a basic matrix multiplication to a figure and could watch the altered figure appear on the screen, albeit slowly. Quite frankly, they loved the course.
This book covers linear algebra before the appearance of formal proofs; I cannot recall seeing a single proof. That coverage is excellent and is focused on how images are created and modified using linear algebra. It is clearly written and illustrated and a tutorial on PostScript appears in an appendix. There is a set of exercises at the end of each chapter and solutions to many of them are included.
A textbook for the modern use of linear algebra as an image creation and modification tool, it is ideal for any math program that wants to cover that material. In my experience, it would be a very popular course, but it cannot be used for any coverage of linear algebra that involves proofs.

Published in Journal of Recreational Mathematics, reprinted with permission

This is a very practical and useful book on linear algebra for students and practitioners in computer graphics and computer science in general. Its practical approach based on its strong visual geometric component will also appeal to students who find other linear algebra books to be a wee bit too dry and theoretical for their tastes.