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I'd say alternative is an unlucky choice of words. I'd rather say geometric algebra (GA) is an extension of linear algebra (LA). In order to really understand GA you need first to firmly understand LA. Then it becomes clear that all that GA does is to turn a Hilbert space into an algebra called a Clifford algebra, and to examine the geometric semantics of the various operations that pop up in the process.

Here are three great sources that helped me to understand GA:

1. https://www.amazon.co.uk/Geometric-Algebra-Computer-Science-...

2. https://www.amazon.co.uk/Linear-Geometric-Algebra-Alan-Macdo...

3. https://www.amazon.co.uk/Algebra-Graduate-Texts-Mathematics-... , pages 749-752

The first source gives great motivation and intuition for GA and its various products. Its mostly coordinate free approach is very refreshing and makes the subject feel exciting and magical. This is also the problem of the book, it's easy to end up confused and disoriented after working through it for a while. The second source is great because it grounds GA firmly on LA, and makes everything very clear and precise. The third source gives a short and concise definition of what a Clifford algebra is.

⬐ klodolph

My personal recommendation for a book on geometric algebra is the one by Hestenes, New Foundations for Classical Mechanics (https://www.amazon.com/dp/0792355148/). I was disappointed by Geometric Algebra for Computer Science and I recently got rid of my copy when I moved to a new apartment, but I have a mathematics background and tend to prefer denser books.

I would say that "alternative" is a viable word here. Yes, you'll need a foundation in linear algebra to understand geometric algebra, but our classes and books on linear algebra go beyond what is necessary for understanding geometric algebra and introduce concepts (like the cross product) which have more natural equivalents in geometric algebra. I'm not even convinced that it's necessary to have a good understanding of matrixes in order to work with geometric algebra.

⬐ auggierose

I guess we have to agree to disagree. GA is not an alternative to LA, as LA is the foundation of GA.

The main point of LA is not matrices, but linear operators, dimensionality, linear independence, bases, etc. Matrices flow naturally from that. If all you have been taught in LA is to manipulate matrices, then I can see why you feel about the relationship between LA and GA the way you do.

⬐ klodolph

You're saying things that I agree with 100% which makes me think that there's something missing from my explanation.

I'm not talking about linear algebra as a field of mathematics in some kind of ideal sense here. Yes, obviously, it's a foundation for geometric algebra. You don't need to convince me of that.

However, elementary linear algebra classes don't teach you about linear operators, they teach you about things like matrixes and cross products. In these basic classes, a "vector" is a "thing with X, Y, and Z coordinates". So when you get to physics, you use the cross product to write a formula for magnetic field. You have to remember that the magnetic field is transformed differently from other vectors according to some special rules. And engineers call this stuff "linear algebra". Mathematicians agree that it's linear algebra, but we know that there's a lot more to linear algebra that goes beyond that.

Alternatively, they could calculate the magnetic field using geometric algebra, and express it as a bivector, at which point all of those special rules vanish.

That's why Hestenes's book is called "New Foundations for Classical Mechanics". It's not that linear algebra is not the foundation for geometric algebra. It's that classes taught in colleges which are called "linear algebra" teach you the concepts used by Gibbs and Wilson in the book Vector Analysis, and these concepts don't generalize to different numbers of dimensions. GA does. Maybe the problem here is that we don't have a special name for that field of study which uses cross products, if had a different name for that stuff, say "vector analysis" after the book first appeared in, we wouldn't have a problems saying that "geometric algebra is an alternative to vector analysis".

GA is a nice alternative to the stuff they teach engineers scientists under the "linear algebra" banner.

Another example… look at Stokes' Theorem. The version with differential forms is a nice alternative to the version with just a cross product.

I found Geometric Algebra for Computer Science (Revised Edition): An Object-Oriented Approach to Geometry to be eye-opening.

https://www.amazon.com/Geometric-Algebra-Computer-Science-Re...

⬐ abdullin

Thank you.

I've had good luck with this one: http://www.amazon.com/Geometric-Algebra-Computer-Science-Rev... They definitely explain why the cross product is an ugly hack :)

The authors have a website - geometricalgebra.net - where you can download a program which will display all the diagrams from the book and allow you to manipulate them. You can also render arbitrary low-dimensional geometric algebra constructions which helps tremendously with improving your intuition about how things work. I'd say it's at a middle ground of mathematical sophistication - it's a good mix of proofs and practical usage. It's almost entirely dedicated to applying geometric algebra to computer graphics, so you won't get as much out of it if you're interested in applications to physics, or just pure mathematics.

If nothing else, I finally know what a quaternion really is after going through this book.

Geometric algebra: http://www.amazon.com/Geometric-Algebra-Computer-Science-Rev...

It is to linear algebra what Lisp is to assembly.