Angle = Volume

This is trippy, and profound.

The determinant — which tells you the change in size after a matrix transformation 𝓜 — is just an Instance of the Alternating Multilinear Map.

(Alternating meaning it goes + − + − + − + − ……. Multilinear meaning linear in every term, ceteris paribus:

begin{matrix} a ; f(cdots  blacksquare  cdots) + b ; f( cdots blacksquare cdots) \ = shortparallel |  | \ f( cdots a  blacksquare + b  blacksquare cdots) end{matrix}    \ \ qquad footnotesize{bullet f text{ is the multilinear mapping}} \ qquad bullet a, b in text{the underlying number corpus } mathbb{K} \ qquad bullet text{above holds for any term } blacksquare text{ (if done one-at-a-time)} )

 

Now we tripThe inner product — which tells you the “angle” between 2 things, in a super abstract sense — is also an instantiation of the Alternating Multilinear Map.

In conclusion, mathematics proves that Size is the same kind of thing as Angle

Say whaaaaaat? I’m going to go get high now and watch Koyaanaasqatsi.

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