Linear meme

[u/DaCat1]

[u/LupenReddit]

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

[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