Page 401 - Advanced Linear Algebra
P. 401
Tensor Products 385
Theorem 14.13 Let =Á Ã Á = and > be vector spaces over - . Then the
mediating morphism map
B¢ hom
²= Á Ã Á =Â > ³ ¦ ²= n Ä n =Á > ³
defined by the fact that is the unique mediating morphism for is an
isomorphism. Thus,
B
hom²= ÁÃÁ= Â> ³ ²= n Ä n = Á> ³
Moreover, if all vector spaces are finite-dimensional, then
dim hom = ² Á Ã Á = Â > ³ µ ~ dim > ² ³ h dim = ² ³
´
~
Theorem 14.8 and its corollary can also be extended.
Theorem 14.14 The linear transformation
Z
Z
Z
Z
B²< Á < ³ n Ä n B²<Á <³ ¦ B²< n Ä n <Á < n Ä n <³
¢
defined by
²
n Än ³²" nÄn" ³ ~ " n Än "
is an embedding and is an isomorphism if all vector spaces are finite-
nÄn of linear transformations is
dimensional. Thus, the tensor product
(via this embedding ) a linear transformation on tensor products. Two important
special cases of this are
i
i
< n Än< Æ ²< nÄn< ³ i
where
² nÄn ³²" nÄn" ³ ~ ²" ³Ä ²" ³
and
i
i
<n Ä n <n = ÆB ²< n Ä n < Á = ³
where
² nÄn n#³²" n Än" ³ ~ ²" ³Ä ²" ³#
Tensor Spaces
Let be a finite-dimensional vector space. For nonnegative integers and ,
=
the tensor product
