What's really nice about this definition is that you don't need the matrix representation of the linear function in order to calculate the determinant of the function.
This shows us how powerful geometric algebra is. In linear algebra, if you want to prove that det(AB) = det(A)det(B) you prove that A can be written as multiplication of elementary matrices, then prove that det(EM) = det(E)det(M) for any elementary matrix E. That is complicated.
@sudgylacmoe , if you consider that the exterior product as an alternating multilinear form which is isomorphic to the dual vector space, and that some people call elements of such dual vector space covectors, i think you'll find it cool that they work just like psuedo vectors! this helped me find that the psuedovectors in 2D is just vectors. i'll the throwing psuedovectors out of my vocabulary because "covector" makes more sense to me, possibly because of my knowledge of category theory.
Not directly related, but any advice on resources or considerations for a projective version of Spacetime Algebra? I was trying to sus out how that all might work on a whiteboard but I only got so far.
@@sudgylacmoe Good to know I wasn't just missing things on my end, I also haven't seen much about it. I have this intuition that light-like lines in spacetime are functioning like "lines at infinity" in some sense, which really intrigues me for some reason.
@@sudgylacmoeGoto (Xylyxylyx UA-cam Channel n you'll c break down uf SPACETIME algebra connections to ya question ! He has over 23 chapters on it ! Checkout da playlist QED Geometric Algebra it's based on MIKOWSKI spacetime deep questions n answers
@@sudgylacmoealso DR DAVID HESTENES specializes books n research n Spacetime algebra , Geometric Calculus & Geometric algebra and been doin it for over 30 years
i think that if you could understand this short (or any of the geometric algebra shorts), u definitely already understand determinants, and dont think they are complicated
Can you simplify vector analysis identities (eg. Div of Grad, Rot of Grad, etc.)? Also, could we bring Kronecker's delta and Levi-civita symbol? Or that would be much of a stretch?
Divergence and curl and all that can be expressed in GA using the inner and outer products, and with them those results generalize to arbitrary dimension. As for the Kronecker delta and the Levi-Civita symbol, the issue there is that they are usually used with components, and GA tries to not use components. You could still use them, but you would lose many of the benefits to GA. You usually can get away without using the Kronecker delta or the Levi-Civita symbol by using the inner and outer products anyway.
@@sudgylacmoe I get it, but at same time, I think it would be nice to abstract a little more, since some multivectors get lenghty, and some operations are repetitive at hand, but well it's notation is still expanding/changing and someone may solve these issues
@@linuxp00 There is also formalism of differential forms, using Hodge duality to go between them and multivectors if necessary. But one way or another, there will be exactly the same number of coordinates one needs to do concrete calculations with. At least for each exterior power by itself. But GA is neat in that you can bunch all those powers together and still have “uniform calculations”. And also there _can be_ computational shortcuts which end users of a GA library don’t need to be aware of.
@@sudgylacmoejust one more thing, linear transforms can be decomposed into a sum of tensor products of basis vectors and basis covectors, so can you do geometric algebra with covectors too? Are these fields really that incompatable? I read as far as the first math stackexchange result on google haha
There are some similarities between covectors and pseudovectors in GA, but I'm not quite sure what the tensor product would look like in that situation. I haven't really looked into this much myself.
You should make some videos about GA ties to differential forms
This would be amazing
I already asked about Geometric Calculus, but he said he's not an expert in it.
What's really nice about this definition is that you don't need the matrix representation of the linear function in order to calculate the determinant of the function.
Idk why i love this kind of videos in which i don't understand a shit but still continues to watching in case i'll understand something XD
This shows us how powerful geometric algebra is. In linear algebra, if you want to prove that
det(AB) = det(A)det(B)
you prove that A can be written as multiplication of elementary matrices, then prove that det(EM) = det(E)det(M) for any elementary matrix E.
That is complicated.
Praise be Geometric Algebra!
@sudgylacmoe , if you consider that the exterior product as an alternating multilinear form which is isomorphic to the dual vector space, and that some people call elements of such dual vector space covectors, i think you'll find it cool that they work just like psuedo vectors! this helped me find that the psuedovectors in 2D is just vectors. i'll the throwing psuedovectors out of my vocabulary because "covector" makes more sense to me, possibly because of my knowledge of category theory.
You love geometric algebra, don't you? Hehe
algebrakisser
hahahaha
@@jamesyeung3286
Not directly related, but any advice on resources or considerations for a projective version of Spacetime Algebra? I was trying to sus out how that all might work on a whiteboard but I only got so far.
This is an active area of research and I haven't heard of any actual papers that have been published yet.
@@sudgylacmoe Good to know I wasn't just missing things on my end, I also haven't seen much about it. I have this intuition that light-like lines in spacetime are functioning like "lines at infinity" in some sense, which really intrigues me for some reason.
I've already heard people say "Point at eternity"
@@sudgylacmoeGoto (Xylyxylyx UA-cam Channel n you'll c break down uf SPACETIME algebra connections to ya question ! He has over 23 chapters on it ! Checkout da playlist QED Geometric Algebra it's based on MIKOWSKI spacetime deep questions n answers
@@sudgylacmoealso DR DAVID HESTENES specializes books n research n Spacetime algebra , Geometric Calculus & Geometric algebra and been doin it for over 30 years
i think that if you could understand this short (or any of the geometric algebra shorts), u definitely already understand determinants, and dont think they are complicated
Can you simplify vector analysis identities (eg. Div of Grad, Rot of Grad, etc.)? Also, could we bring Kronecker's delta and Levi-civita symbol? Or that would be much of a stretch?
Divergence and curl and all that can be expressed in GA using the inner and outer products, and with them those results generalize to arbitrary dimension. As for the Kronecker delta and the Levi-Civita symbol, the issue there is that they are usually used with components, and GA tries to not use components. You could still use them, but you would lose many of the benefits to GA. You usually can get away without using the Kronecker delta or the Levi-Civita symbol by using the inner and outer products anyway.
@@sudgylacmoe I get it, but at same time, I think it would be nice to abstract a little more, since some multivectors get lenghty, and some operations are repetitive at hand, but well it's notation is still expanding/changing and someone may solve these issues
@@linuxp00 There is also formalism of differential forms, using Hodge duality to go between them and multivectors if necessary. But one way or another, there will be exactly the same number of coordinates one needs to do concrete calculations with. At least for each exterior power by itself. But GA is neat in that you can bunch all those powers together and still have “uniform calculations”. And also there _can be_ computational shortcuts which end users of a GA library don’t need to be aware of.
How do you act the linear transform on the bivector?
By using the outermorphism. You use f(e12) = f(e1 ∧ e2) = f(e1) ∧ f(e2).
@@sudgylacmoethat makes sense, thanks!
@@sudgylacmoejust one more thing, linear transforms can be decomposed into a sum of tensor products of basis vectors and basis covectors, so can you do geometric algebra with covectors too? Are these fields really that incompatable? I read as far as the first math stackexchange result on google haha
There are some similarities between covectors and pseudovectors in GA, but I'm not quite sure what the tensor product would look like in that situation. I haven't really looked into this much myself.
Oh I forgot all those things...😢
Bro I have multiple degrees and I have no idea wtf ur talking about
proof more degrees doesn't mean understanding understanding every field/observation made in mathematics? lol