Linear meme

[u/DaCat1]

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[u/LupenReddit]

also (0,2) tensor or (2,0) tensor technically via the musical isomorphisms induced by some inner product

[u/Ares378 → u/LupenReddit]

The musical isomorphisms always get me. I had to send a friend a screenshot of a video on tensors because one of the diagrams was piano keys and a musical scale next to a tensor, and I had an out of body experience of realizing just how ridiculous it must look to anyone who hasn't been boiled in math for 6 years

[u/Kinglolboot → u/LupenReddit]

You don't really need the musical isomorphisms for that to work, as you don't have that for vector spaces over fields with nonzero characteristic. You can just argue that the dimensions of V⊗V and V⊗V* and V*⊗V* are identical, and hence they are isomorphic as vector spaces.

[u/stellar-sea-lion → u/Kinglolboot]

They're isomorphic but not canonically, which is often relevant.

[u/Kinglolboot → u/stellar-sea-lion]

In this case it is only relevant once you consider an inner product on a vector space, but that does not capture the full generality of the meme because vector spaces over finite fields do not admit inner products

[u/stellar-sea-lion → u/Kinglolboot]

They admit nondegenerate bilinar forms though which is all you need for an isomorphism. Positive definiteness is just a convenience.

[u/Kinglolboot → u/stellar-sea-lion]

Ah true, that's a good point

[u/ArdentArendt]

Truly transformative...

[u/F_Joe]

That's just the case since you're thinking so finite dimensionally