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Tensor product

A function of two vector spaces, U and V, which returns the space of linear maps from V’s dual to U.
Tensor product has natural symmetry in interchange of U and V and it produces an associative “multiplication” on vector spaces.
Wrinting * for tensor product, we can map UxV to U*V via: (u,v) maps to that linear map which takes any w in V’s dual to u times w’s action on v. We call this linear map u*v. One can then show that
u * v + u * x = u * (v+x) u * v + t * v = (u+t) * v and hu * v = h(u * v) = u * hv
ie, the mapping respects linearity: whence any bilinear map from UxV (to wherever) may be factorised via this mapping. This gives us the degree of natural symmetry in swapping U and V. By rolling it up to multilinear maps from products of several vector spaces, we can get to the natural associative “multiplication” on vector spaces.
When all the vector spaces are the same, permutation of the factors doesn’t change the space and so constitutes an automorphism. These permutation-induced iso-auto-morphisms form a group which is a model of the group of permutations.


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