[catnaroek]

> OK so what is this Geometric Algebra? It’s an alternative to linear algebra.

No. Geometric algebra is a use case of linear algebra. How can it be an alternative?

> Before I tell you how to actually evaluate the wedge product, I first have to tell you the properties that it has:

> 1. It’s anti-commutative: a \wedge b = -b \wedge a

> 2. The wedge product of a vector with itself is 0: a \wedge a = 0

Redundant information. The latter follows from the former.

[klodolph]

It's definitely an alternative in the sense that it gives you an alternative framework for concepts that are taught under the banner of linear algebra in school. For example, it gives an alternative construction for quaternions as a subalgebra, and it gives the exterior product as an alternative to the cross product.

[auggierose]

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.

[madhadron]

No, it's not an alternative. You're conflating the grab bag of topics in an undergraduate linear algebra class with the subject of linear algebra.

Linear algebra is the study of linear operators on vector spaces over fields (a special case of modules over rings). Some vector spaces are inner product spaces, but most are not.

Exterior algebra is an example of multilinear algebra. Clifford (or geometric) algebras are constructed as an algebra over a vector space. Quaternions come via a different route: instead of constructing a multilinear algebra, they are one of a handful of very special algebras like C and the octonions.

[catnaroek]

Exterior algebras themselves are defined in terms of standard linear algebra concepts (tensor algebras, ideals, quotients). I don't see how you can claim something is an alternative to linear algebra if you need to know linear algebra to define it, nevermind use it.

[Retra]

Left is an alternative to right. Right is the opposite of left. They are alternatives, yet defined in terms of each other. You can't know one without the other.

Doesn't seem contradictory to me...

[sebastos]

> Redundant information. The latter follows from the former.

It's a blog post, not a paper. If I give you a brand new operator ^ that takes an arbitrary object a and another object b and tell you that it has anti-commutativity, it's not a given that a ^ a = 0. So it's useful to highlight that this IS the case when a and b are the vectors of GA, for intuition's sake. a = -a iff a is the zero vector. What if our objects are rotations or something though? R = -R does not imply R = the zero rotation.

[catnaroek]

^ is the multiplication operator of an algebra, though.