Page 398 - Advanced Linear Algebra

382 Advanced Linear Algebra

² n #³ ~ ² n #³
~ ² k ³²#³
=
~ ² > k ³²#³

~² n #³
~ n #

~² 2 n ³² n #³
and so ~ . n 2
Theorem 14.11 Let and > be -spaces, with 2 -extension maps = and
=
-
( ) - > , respectively. See Figure 14.7. Then for any -linear map ¢ = ¦ > , the
map 2 n ¢ 2n = ¦ 2n > is the unique 2 -linear map that makes the
diagram in Figure 14.7 commute, that is, for which
~ ² n 2 k ³
k
Multilinear Maps and Iterated Tensor Products

The tensor product operation can easily be extended to more than two vector
spaces. We begin with the extension of the concept of bilinearity.

Definition If =Á Ã Á = and > are vector spaces over - , a function
¢ = d Ä d = ¦ > is said to be multilinear if it is linear in each coordinate

separately, that is, if
² " Á Ã Á " Á c # b # Z Á " Á Ã b Á " ³
~ ² " Á Ã Á " Á # c Á " Á Ã b Á " ³ b ² " Á Ã Á " Á # c Z Á " Á Ã b Á " ³

for all ~ Á Ã Á . A multilinear function of variables is also referred to as
an . The set of all -linear functions as defined above will be
-linear function
denoted by hom²= ÁÃÁ= Â> ³ . A multilinear function from = d Ä d = to

the base field is called a multilinear form or -form .
-

Example 14.7
)
(
1 If is an algebra, then the product map ¢ ( d Ä d ( ¦ ( defined by
is -linear.
² ÁÃÁ ³ ~ Ä

) ¢ ¦ - is an -linear form on the columns

2 The determinant function det C
of the matrices in C .
The tensor product is defined via its universal property.
be the cartesian
Definition As pictured in Figure 14.8, let =d Ä d =
-
product of vector spaces over . A pair ²;Á !¢ = d Ä d = ¦ ;³ is universal

for multilinearity if for every multilinear map ¢ = d Ä d = ¦ > , there is

a unique linear transformation ¢; ¦ > for which