Page 377 - Advanced Linear Algebra

Tensor Products 361

² Á ³ ~
is bilinear, that is, multiplication is linear in each position.

Example 14.5 (Evaluation is bilinear ) If and > are vector spaces, then the
=
evaluation map B¢²= Á > ³ d = ¦ > defined by
² Á #³ ~ #
i
is bilinear. In particular, the evaluation map ¢= d = ¦ - defined by
i
² Á #³ ~ # is a bilinear form on = d = .
Example 14.6 If and > are vector spaces, and  = i and  > i , then the
=
product map ¢= d > ¦ - defined by
²#Á $³ ~ ²#³ ²$³
i
i

is bilinear. Dually, if #= and $> , then the map ¢ = d > ¦ -
defined by
² Á ³ ~ ²#³ ²$³
is bilinear.

It is precisely the tensor product that will allow us to generalize the previous
B
B

example. In particular, if ²<Á > ³ and ²= Á > ³ , then we would like
to consider a “product” map ¢< d = ¦ > defined by
²"Á #³ ~ ²"³ ? ²#³
The tensor product n is just the thing to replace the question mark, because it
has the desired bilinearity property, as we will see. In fact, the tensor product is
bilinear and nothing else, so it is exactly what we need!
Tensor Products
Let and be vector spaces. Our guide for the definition of the tensor product
<
=
<n = will be the desire to have a universal property for bilinear functions, as
measured by linearity. Referring to Figure 14.4, we want to define a vector
space and a bilinear map ! ¢ < d = ¦ ; so that any bilinear map with
;

!
domain <d = can be factored through . Intuitively speaking, is the most
!
“general” or “universal” bilinear map with domain <d = : It is bilinear and
nothing more.