Page 379 - Advanced Linear Algebra

Tensor Products 363

As we will see, there are other, more constructive ways to define the tensor
product. Since we have adopted the universal pair definition, the other ways to
define tensor product are, for us, constructions rather than definitions. Let us
examine some of these constructions.

Construction I: Intuitive but Not Coordinate Free
The universal property for bilinearity captures the essence of bilinearity and the
tensor map is the most “general” bilinear function on <d = . To see how this
universality can be achieved in a constructive manner, let ¸ “  0¹ be a basis

!
<
for and let ¸ “  1 ¹ be a basis for . Then a bilinear map on < d = is
=
uniquely determined by assigning arbitrary values to the “basis” pairs ² Á ³

and extending by bilinearity, that is, if "~ and # ~ , then
5
!²"Á #³ ~ !4 Á ~ !² Á ³

Now, the tensor map , being the most general bilinear map, must do this and
!
nothing more. To achieve this goal, we define the tensor map on the pairs
!
² Á ³ in such a way that the images !² Á ³ do not interact , and then extend

by bilinearity.
In particular, for each ordered pair ² Á ³ , we invent a new formal symbol,

, and define to be the vector space with basis
written n ;
: ~¸ n “ 0Á 1¹

and extending by

The tensor map is defined by setting !² Á ³ ~ n

bilinearity. Thus,
5

!²"Á #³ ~ !4 Á ~ ² n ³

=
<
To see that the pair ²;Á!³ is the tensor product of and , if ¢< d = ¦ > is
bilinear, the universality condition ~ k ! is equivalent to
² n ³ ~ ² Á ³

which does indeed uniquely define a linear map ¢; ¦ > . Hence, ²;Á!³ has
the universal property for bilinearity and so we can write ;~ < n = and refer
to as the tensor map.
!
;
Note that while the set : ~¸ n ¹ is a basis for (by definition), the set

¸" n # “"<Á # = ¹
of decomposable tensors spans , but is not linearly independent. This does
;
cause some initial confusion during the learning process. For example, one
cannot define a linear map on <n = by assigning values arbitrarily to the
is
decomposable tensors, nor is it always easy to tell when a tensor "n #