Linear and Geometric Algebra (Geometric Algebra & Calculus)

Alan Macdonald's text is an excellent resource if you are just beginning the study of geometric algebra and would like to learn or review traditional linear algebra in the process. The clarity and evenness of the writing, as well as the originality of presentation that is evident throughout this text, suggest that the author has been successful as a mathematics teacher in the undergraduate classroom. This carefully crafted text is ideal for anyone learning geometric algebra in relative isolation, which I suspect will be the case for many readers, given that geometric algebra is not a standard part of the undergraduate mathematics, physics, or engineering curriculum.Like most everyone else, I first became aware of geometric algebra through David Hestenes: his American Journal of Physics articles, his books, and the many materials available at his web site, all of which I can recommend. I've also spent considerable time with the geometric algebra book by Doran and Lasenby, as well the book by Dorst, Fontijne, and Mann. The aforementioned books can help you understand why it might be worth your while to learn geometric algebra. Should you decide geometric algebra is worth your while, and furthermore, decide to develop some pencil-and-paper proficiency with it, I recommend Macdonald's book as a great way to get started. It won't help you discover new applications of geometric algebra, but it will give you the mathematical background and confidence you need to move on to the more difficult books and articles with applications to science/engineering.Macdonald writes in a consistently friendly, but serious, voice that suggests he cares whether or not a reader understands the reasoning behind proofs and appreciates the significance of the results obtained from them. This is not a high-powered mathematics monograph for graduate students and researchers--this a book for first-time learners. Macdonald does not show off how much more he knows about this topic than you do. The linear algebra material in this book was well known to me from my undergraduate courses, and I use most of it regularly in physics; still I (re)learned a great deal about the nature of mathematical proof that was helpful later in the geometric algebra half of the book. It was great to have a consistent voice throughout both sections. I did a number of the 200+ exercises/problems, and felt the preceding sections had just what I needed to get an exercise/problem done.There are remarkably few typographical errors--the few I did find were minor and did not obscure the meaning of the text. Some books on geometric algebra are full of typographical errors, to the point that I could imagine readers giving up in exasperation; the editors and authors of those books did not do their jobs. Macdonald's book has no such defect. Amazon allows you to look at a great number of pages from Macdonald's book on-line (today I could look at 90+ pages on-line). Read a section or two--note the many helpful diagrams.This is not a book about physics, yet there are a few informative asides pertaining to physics, an example of which is a set of photographs showing an everyday situation where a rotation by 720 degrees is required to return a system to its original state. Also, as mentioned in the book's preface, some standard techniques in linear algebra are not covered (one uses a computer to carry out these calculations anyway) because the digressions necessary to cover them would have drawn attention away from making the transition to geometric algebra. If you only want to learn linear algebra in order to crunch numbers, this probably isn't your book. On the other hand, if money motivates you, the section on the $25,000,000,000 eigenvalue just might make this book a profitable purchase.I hope Macdonald will find time to write a comparable book on geometric calculus, which is what I really need and want to know, but have found difficult to learn from other sources.

I have really enjoyed reading and re-reading this book. I wish I could have learned Geometric Algebra when I was starting out a few decades ago. Not only does Geometric Algebra bring new tools for geometry, it also adds new insight and power to Linear Algebra. I like the way that Professor Macdonald emphasizes an intuitive point of view and practical applications, yet it is mathematics and not tourism. I highly recommend this book, and also the sequel Vector and Geometric Calculus.

Alan Macdonald has prepared for this for a long time. His personal web site at Luther College in Iowa includes an extensive list of his papers. That work shows clearly that he has both the mathematical and physical chops for this task. His detailed survey of geometric algebra and geometric calculus, which can be found on that site, has been worked and reworked since he first developed it in 2002. It is quite thorough in that it covers not just the fundamentals of the algebra, but also incorporates quite a few physical applications for motivation. For example, at this point it's well understood that quantum spin is really a geometric property of the particles themselves. Macdonald's survey covers that clearly and concisely.Here he develops the first undergraduate text to cover the essentials of linear algebra, and its extension to geometric algebra. The terse statements from the above survey are expanded, in this elegant book, into rigorous proofs. Given the care with which that's done, however, it easily rewards those students for whom this is a first introduction to the abstract concepts inherent in linear vector spaces - and the higher dimensional analogues where the multi-vectors of geometrical algebra live.I believe, as Macdonald does, that the geometric interpretation of Clifford algebras, and its extension to geometric calculus "unify, simplify, and generalize vast areas of mathematics". I'd strongly recommend this book to engineering, computer science, and physics teachers. It provides a solid grounding in this important and emerging area of mathematics.

Linear algebra is really a foundational course. If it doesn't explain so much about the world, it explains a very great deal about the way mathematicians think about the world. This book is written at exactly the level to give students a first view into the subject. So, it collects the facts in a very concise and straightforward way. A freshman or sophomore in college (or even a good high school student) could use this as a fine introduction.It was particularly insightful and very necessary for the author to include geometric algebra as a second half of the book. It goes beyond the regular curriculum and gives insight into mathematics that is very useful for computer applications and for general calculation. It is really a way of the future, and students are brought in on the ground floor.Most people would see nothing to object to in this book. I would like to have seen a more detailed and constructive presentation. The author states (correctly) a basis of G^3 without going into detail of why the stated elements do form a basis. How does a beginner know they're linearly independent? (They look independent, isn't that enough????) If he had been willing to be more careful and constructive, I think it would have been a great book. As it is, the book is useful and ought to be considered at many schools, although I think there are certainly much better linear algebra texts (Linear Algebra Done Right, for example). None of the better texts give the huge advantage of including geometric algebra, as they really ought to do.