Linear and Geometric Algebra (Geometric Algebra & Calculus)

I am using this book for self-study and was, at first, disappointed to find that there are no answers to the exercises. However as I have progressed through the book I have realised that Alan Macdonald covers everything you need to know to successfully complete them. The result is that the book imparts a thorough understanding of the material it covers and it's fun! The coverage of topics is excellent. The explanations are clear, thete are helpful diagrams, it is an excellent book and I thoroughly recommend it.

Fantastic read. I've been interested in Geometric Algebra for a long time and this is an excellent and clear introduction to the subject. Linear algebra concepts are gradually introduced, studied and then generalized via GA - it's actually a truly exciting book to read (I'd say especially if you've read a bit of GA in the past and know where things are leading!). There are a good selection of problems that are worth coming back to multiple times but it may be a little hard for self study unless you're comfortable with the style of text. For self study this probably works best as a companion to a more traditional linear algebra book with worked examples. For the size of the book and cost it covers a good selection of topics, not really a criticism but it does leave you wanting to learn more! So much in fact that I've just ordered the second book, Vector and Geometric Calculus by the same author.

This is the first undergraduate textbook I have come across and read on linear algebra that uses geometric algebra in an integrated and natural way, one which will be both eye-opening and enormously benficial to students and other readers who have not seen it before. I would echo David Hestenes' positive comments on the cover of the book - Hestenes is the acknowledged creator of geometric algebra in the form it appears in the book - in particular, that it uncovers some of the gems of geometric algebra. However, as a word of warning, purchasors and readers should not expect what the book neither claims nor attempts to deliver: e.g. any of Hestenes' geometric calculus on vector manifolds. That is, it contains some of the gems but by no means all and, for that, one must look elsewhere.

Very good quality

A completely new view on vectors providing generalizations of vectors, extensions to the vector notion into higher dimensions.

I used this book to construct my first understanding of the subject. I would recomend it to any student. It is an excellent guide into the subject. The binding quality and paper quality are also excellent.

Excellent from both educational and content viewpoints. The student, either undergraduate or graduate, is driven with simplicity to the book objectives with a number of meaningful exercises - important for the comprehension of the topic.

Differential formsや最近の数理物理関係の本で使用されるExterior productの背景が知りたくてこの本を購入しましたが、結果としてGAを勉強することができました。ベクトル空間の考え方が多方面に利用できることがわかりました。GAについては、コンピューターグラフィクスをはじめ、物理・工学の色々な問題に応用できそうなので、他にもGAの応用に関する本を読んでみるつもりです。（この本には実際の応用例の詳細の記述はない）

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.

This book has a unique approach. It starts by showing the most common linear algebra topics in a clever, direct and compact way. I find this approach very useful, because its main purpose is to teach the basic concepts, without going into many details (such as how to compute inverse matrices and so) which would otherwise obscure the actual teaching concept. Once the main linear algebra topics are covered, the author ventures into the beautiful universe of geometric algebra using, of course, the same approach. The result is an unique book, which somehow represents a guide on how to dive into geometric algebra starting from "familiar" or more common linear algebra topics. My only concern about this book is the lack of a solutions to the exercises and problems. Maybe the former are not strongly required, but for someone who is learning GA on his own (without a teacher or supervisor, capable of answering questions) the lack of solutions for the problems represent a drawback. On the other hand the software tools provided by the author on his website are actually quite useful. In general, I recommend this book.