Page 378 - Advanced Linear Algebra

362 Advanced Linear Algebra

t bilinear
U V T

W linear
f bilinear
W

Figure 14.4
-
Definition Let <d = be the cartesian product of two vector spaces over . Let
I ~ ² Vect - ³ . Let
I ~
< ¸hom - ²<Á= Â> ³ “ >  ¹
>
be the family of all bilinear maps from <d = to any vector space > . The
measuring family is the family of all linear transformations.
>

A pair ²;Á!¢< d = ¦ ;³ is universal for bilinearity if it is universal for
²Á<> ³, that is, if for every bilinear map ¢ < d = ¦ > , there is a unique
linear transformation ¢; ¦ > for which

~ k !
The map is called the mediating morphism for .

We can now define the tensor product via this universal property.

Definition Let and be vector spaces over a field . Any universal pair
=
<
-
²;Á!¢< d = ¦ ;³ for bilinearity is called a tensor product of < and . The
=
vector space is denoted by < n = and sometimes referred to by itself as the
;
tensor product. The map is called the tensor map and the elements of < n =
!
are called tensors .
It is customary to use the symbol n to denote the image of any ordered pair
²"Á #³ under the tensor map, that is,
"n# ~ !²"Á #³
for any "< and # = . A tensor of the form " n # is said to be
decomposable, that is, the decomposable tensors are the images under the
tensor map.

Since universal pairs are unique up to isomorphism, we may refer to “the”
tensor product of vector spaces. Note also that the tensor product n is not a
product in the sense of a binary operation on a set. In fact, even when =~ < ,
the tensor product "n" is not in , but rather in < n < .
<