Linear and Geometric Algebra (Geometric Algebra & Calculus)

First, a disclaimer of interest. I'm the "Isaac To" Alan referred to in the preface of his book. I find the approach of the book very enjoyable. Although I read the book with a full understanding of linear algebra and with some earlier reading of introductions to geometric algebra, the book actually requires neither as a prerequisite. Although the book targets students of university courses, good secondary students should have no issue with the book if assisted by a knowledgeable tutor. I've tested this statement by giving the book to my son at the age of 14, who successfully understood most of the contents with only minor occasional calls for my assistance. Most books about linear algebra would let you focus on numbers and solving systems of linear equations, before you will ever get into the realm of linear algebra. Instead, Alan would have you think about the vector-like objects around you. Then you are asked to reason about them directly, forming linear combinations, selecting bases, understanding dimensions and ranks, and so on. The idea that you can use vectors to solve a system of linear equations is left much later as an application. The procedures to find an inverse of a square matrix is completely skipped except for the 2x2 case, delegating the task to computer programs. The book let you work with numbers only occasionally, to let you confirm that you know the concepts that are introduced in the book. Instead, there are lots of exercises which are conceptual. Even some of the actual contents like theorems and lemmas are delegated as such exercises. I think this is a little controversial, because if you are not mindful about the words of the book, it is easy to find a needed lemma in an exercise that you don't know how to work out. They are all easy (if you look at the problems in the right perspective), however, so such omissions actually promote your understanding of the materials by confirming that you understand what you read. My feeling is that readers would gain a lot more insight on how vectors and transformations work together than a traditional approach when learning linear algebra. Another interesting aspect of the book is of course to treat geometric algebra as a first-class citizen rather than an add-on of the theory. Geometric algebra is introduced in the book before linear transformation is introduced. This means that readers will have more tools in their disposal when working with linear transformations. As an example, the discussion of determinants is delayed so much that the readers already know about outermorphisms. And determinants are introduced first for linear transformation, before for matrices. Again this may be slightly controversial: many readers may not have the chance to reach the later chapters of the book. Delaying an important concept so far might mean that some readers will never have a chance to learn it. On the other hand, because it is introduced so late in the game, the concept of determinant becomes very intuitive, whereas in regular linear algebra books it is mostly just a tool for computations. Treatments of other topics are also very interesting, like (1) matrix transpose is introduced only after the adjoint of linear transformation as its representation, (2), transformations are treated as more important citizen than matrices and the book talks about special transformation rather than special matrices, (3) an asymmetric form is chosen for the general definition of the inner product of multivectors, and so on. They all contribute to a great read for me.

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.

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.

REVIEW: Linear and Geometric Algebra (Geometric Algebra & Calculus) DATE: 4-Apr-26 As I write this review I have about 30 hours in this book, I'm on chapter 5.1 "Oriented Areas" pp. 73/200 About an hour per night. This book already has a 5 star reputation and is "endorsed" by David Hestenes himself. I will just add that this is VERY approachable for "self teaching" if you already have a technical degree with calc though ODE and a semester doing vector calc. If you want to expand your tool set, this is THE book to start with. I would add that using it can make you feel like you are missing something because the manipulations are far less tedious than traditional vectors. Has numerous examples and thought provoking problem sets. There are no "answers in the back of the book" but at this level you should not really need that, Right? If you are a undergrad or grad student with LA already in your tool set and you are thinking about learning this tool set: Buy This Book! I am looking forward to taking on Dr. MacDonald's "Vector and Geometric Calculus" work later this spring and I honestly wish I had studied this 30 years ago.

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.

Macdonald introduces the geometric algebra in a clear and logical manner. The link between the geometric algebra framework, with its focus on coordinate-free (= basis independent) expressions, and the traditional (coordinate-dependent) linear algebra is clearly presented. Macdonald's book is carefully written and I haven't discovered any significant typos in the second edition. A lot of ground is covered in such a 'thin' book. Although the book contains only about 200 pages, it is densely packed and the reader is supposed to get his/her hands dirty by doing the exercises and problems. These exercises and problems typically ask you to complete proofs of theorems stated in the text or to derive/prove supplementary theorems/lemmas. In addition there a many exercises dealing with practical applications of linear algebra. A great exposition is given on the widely used linear transformations in 3D: the projections, rotations, reflections and their generalizations to higher dimensions.

Poor, often confusing, presentation of material, few worked examples, no applications to justify the effort to decipher this mess.

I'm no expert on linear algebra, rather just an older student who sees a lot of connections to various interesting corners of mathematics. I'm working my way through learning linear algebra, and am using a couple of texts -also Strang - to get a rounded view. Macdonald's text could use problems with solutions but the approach is different enough to make it a great second text to work from. I would say the strengths of this book decidedly outweigh the weaknesses. It is not for a casual read. It demands work. I'm totally glad I'm using it.