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tensor product
mathematics 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 bilinearmap 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.