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What other way is there around avoiding polar vs axial vectors looking the same but behaving differently?

Or normal vs tangent vectors transforming differently?

Those concepts are from exterior algebra which GA is built on. You do not need GA (the geometric product, etc) for them.

By the way, I love your blog post https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm...

I didn't understand this part:

> I strongly believe that if GA would make this distinction they would lose a lot fewer people. It is a completely interesting and useful thing to talk about “a representation of a particular class of operations that makes composition and inversion easy”, and completely offputting when you blur the distinction between operators and geometric objects themselves, and write every operation in terms of the geometric product when only a few of them are really compositions of operators.

I can't tell what "a few of them" refers to. What is this potential distinction between operators and geometric objects? Sounds like the the distinction between a group action and a group object?

I am willing to believe that GA is an unnecessary renaming of other simpler things, and also that it has these kind of culty vibes, but I'm focusing on the claim that (I understood as) "unifying the operators and geometric objects" is a bad feature rather than a good feature.

So, I've amended the article some since posting it because the main objection I got from a few GA enthusiasts was this idea of the GP as being used to compose operators. Which I had barely noticed the importance of because it's so hard to identify in the texts! Although once they mentioned it I began to appreciate that, when the GP works and is useful, this is why. So I changed things to address that point more directly... but I didn't really find the energy to do a perfect job of it (like researching all the examples I would need to make the point more clearly), and it's a bit muddy at the moment. Like at the spot you pointed out. I need to figure out a clearer way to say this stuff...

What I am getting at is that if you go read, say, the Doran/Lasenby book, they start out talking about multivectors for areas and volumes and etc---and they do all this with the GP. Which makes no sense! Ever calculation they do leaves you think "huh?" The GP makes no sense at all if you're talking about units of length, area, volume, etc. Its transformation laws, its composition laws... you end up having to undo it all afterwards with a bunch of janky other operations.

But if you talk about the GP for composing reflections to make rotations, it's fine, that makes sense. I just really want this distinction to be made more clearly. I'm only interested in the GP when it corresponds to an explicit geometric operation. Nobody makes this distinction as clear as I want; I hope to eventually find a really sound version of the argument and then write it out as another article.

Roughly speaking it's equivalent to conflating the sense of a complex number as a vector with a complex number as an operator on vectors. Yes, they're isomorphic, but given a vector in R^2 there's no intrinsic sense in which you should be able to interpret it as also being the operation of multiplying by r e^(iθ) on other vectors. Pretending like they're the same thing is just bewildering: that identification between vectors and operations should be something you have to explicitly construct. For starters, if you change bases for (x,y) the vector should rotate but the rotation operation shouldn't change. That sort of thing. GA is making this same confusion but on a larger scale.

elengyel 654 days ago

My position is that the geometric product and antiproduct are good for one thing, performing transformations with sandwich products q ⟑ p ⟑ q̃ or q ⟇ p ⟇ q̰ and composing those transformations. Literally everything else (join, meet, contraction, expansion, projection, inner product, norm, ...) can and should be done in the exterior algebra without any geometric products.

ajkjk 653 days ago

agree but I am still trying to grok the divine truth as to why exactly that is. What's up with the sandwich products? Why do they work? I guess it is like a change-of-basis for a matrix (PAP^{-1)) but I still don't quite see why, and why it works as a change-of-basis on multivectors, not just vectors.

That's fine. That's still a step beyond just "linear algebra".

Oh yes, strongly agree. They should be taught in linear algebra really.

- https://en.wikipedia.org/wiki/Exterior_algebra

Happy to have agreement that this goes beyond "traditional linear algebra".

But, pedagogical treatment is a separate question from what is linear algebra.

Should these all be the same wikipedia page?

- https://en.wikipedia.org/wiki/Multilinear_algebra

- https://www.georgehart.com/research/multanal.html (okay, not mainstream enough to have a relevant page, but it is extremely relevant in any engineering practice of linear algebra)

Well by "taught in linear algebra" I mean "taught to undergraduates in their first or second year". It is totally bizarre that people learn about determinants, matrix minors, curl, magnetic fields, angular momentum, etc... but not about wedge products, in which all of these things are much easier to understand.