Page 381 - Advanced Linear Algebra

Tensor Products 365

<n = has the form

"n #
as in the previous construction.

The tensor map !¢ <d = ¦ <n = is defined by
!²"Á #³ ~" n # ~²"Á #³ b :

This map is bilinear, since
!² " b #Á $³ ~ ² " b #Á $³ b:
~ ´ ²"Á $³ b ²#Á $³µ b:
~´ ²"Á $³ b :µ b ´ ²#Á $³ b :µ
~ !²"Á $³ b !²#Á $³
and similarly for the second coordinate.

We next prove that the pair ²< n= Á !¢ < d= ¦ < n= ³ is universal for
bilinearity when <n = is defined as a quotient space - <d= °: .

Theorem 14.3 Let and be vector spaces. The pair
=
<
²< n= Á !¢ < d= ¦ < n= ³
is the tensor product of and .
=
<
is the vector space with
Proof. Consider the diagram in Figure 14.5. Here - <d=
basis <d = .
t

j S
U V F U V U V

f V W

W
Figure 14.5

Since
k ²"Á #³ ~ ²"Á #³ ~ ²"Á #³ b : ~ " n # ~ !²"Á #³
we have
!~ k

The universal property of vector spaces described in Example 14.1 implies that