Page 375 - Advanced Linear Algebra

Tensor Products 359

=
2) < ~ linear maps with domain , whose kernels contain 2
3) > ~ linear transformations

Theorem 3.4 says precisely that the pair ²= °2Á ¢ = ¦ = °2³ , where is the

<
>
canonical projection map, has the universal property for as measured by .
Example 14.3 (Direct sums ) Let and be vector spaces over . Let
-
=
<
1) I ~ ² Vect - ³
2) < ~ ordered pairs ² ¢ < ¦ >Á ¢ = ¦ >³ of linear transformations
3) > ~ linear transformations
Here we have a slight variation on the definition of universal pair: In this case,
< is a family of pairs > and ² Á ³  < of functions. For  , we set
k² Á ³ ~ ² k Á k ³

Then the pair ²< ^ = Á ² Á ³¢ ²<Á = ³ ¦ < ^ = ³ , where
" ~ ²"Á ³ and # ~ ² Á #³

are called the canonical injections , has the universal property for ²Á<> ³ . To
<
see this, observe that for any pair ² Á ³¢ ²<Á = ³ ¦ > in , the condition
² Á ³ ~ k ² Á ³

is equivalent to
² Á ³ ~ ² k Á k ³

or
and ²"Á ³ ~ ²"³ ² Á#³ ~ ²#³
But these conditions define a unique linear transformation ^¢< = ¦ > .

Thus, bases, quotient spaces and direct sums are all examples of universal pairs
and it should be clear from these examples that the notion of universal property
is, well, universal. In fact, it happens that the most useful definition of tensor
product is through a universal property, which we now explore.

Bilinear Maps
The universality that defines tensor products rests on the notion of a bilinear
map.

,
Definition Let <= and > be vector spaces over - . Let < d = be the
cartesian product of and as sets . A set function
=
<
¢ < d = ¦ >