Source : Free On-Line Dictionary of Computing

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.
     
        (1996-09-27)