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.

(1996-09-27)