Page 397 - Advanced Linear Algebra

Tensor Products 381

is bilinear and so there exists a unique -linear map for which

-
²n #³ ~ ²#³
2

It is easy to see that is also -linear, since if  2 , then
n#³µ ~ n#³ ~ ²#³ ~ n#³
´
²
²
²
P
V F K V F
W
f
Y

Figure 14.6
Theorem 14.10 is the key to describing how to extend an -linear map to a -
2
-
linear map. Figure 14.7 shows an -linear map ¢ = ¦ > between -spaces =
-
-
and > . It also shows the 2 -extensions for both spaces, where 2 n = and
2n > are 2-spaces.
W
V W

P V P W

W
K V K W

Figure 14.7

If there is a unique 2 -linear map that makes the diagram in Figure 14.7
commute, then this would be the obvious choice for the extension of the -
-
linear map to a -linear map.
2

Consider the - -linear map ~ ² k ³ ¢ = ¦ 2 n > > into the 2 -space
2n > . Theorem 14.10 implies that there is a unique 2-linear map
¢ 2 n= ¦ 2 n> for which
k = ~
that is,

k = ~ > k
Now, satisfies