Determinants in Geometric Algebra

In this short, I show how outermorphisms can lead to an incredibly simple definition of the determinant. This definition is simple both geometrically and algebraically, and it is incredibly useful.

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
trb

[maxwellmogadam]

You should make some videos about GA ties to differential forms

[Aryan-tmsm]

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

[richardcoppin]

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.

[siarya_math]

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.

[GamingKing-jopy]

@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.

[Jordan-zkwd]

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.

[phygeniuxYTcreations]

You love geometric algebra, don't you? Hehe

[linuxp]

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?

[lih]

How do you act the linear transform on the bivector?

[SuperMaDBrothers]

Bro I have multiple degrees and I have no idea wtf ur talking about

[AsiccAP]

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