Page 395 - Advanced Linear Algebra

Tensor Products 379

Change of Base Field
The tensor product provides a convenient way to extend the base field of a
vector space that is more general than the complexification of a real vector
space, discussed earlier in the book. We refer to a vector space over a field as
-
an -space and write - .
-
=
Actually, there are several approaches to “upgrading” the base field of a vector
space. For instance, suppose that is an extension field of , that is, - ‹ 2 .
2
-
If ¸ ¹ is a basis for = - , then every %  = - has the form

%~

where  - . We can define a 2 -space = 2 simply by taking all formal linear

combinations of the form

%~
2
where . Note that the dimension of 2 2 = as a -space is the same as the
(
-
dimension of - = as an -space. Also, 2 - = is an -space just restrict the scalars
to - and as such, the inclusion map ) ¢ = - ¦ = 2 sending %  = - to
²%³ ~ %  = 2 is an -monomorphism.
-
The approach described in the previous paragraph uses an arbitrarily chosen
and is therefore not coordinate free. However, we can give a
basis for = -
coordinate-free approach using tensor products as follows. Since 2 is a vector
space over , we can form the tensor product
-
>~ 2 n = - -
-
-
It is customary to include the subscript on n - to denote the fact that the
tensor product is taken with respect to the base field . (All relevant maps are
-
=
- --bilinear and -linear.) However, since - is not a 2 -space, the only tensor
-
product of 2 and = - that makes sense is the -tensor product and so we will
drop the subscript .
-
The tensor product > is an -space by definition of tensor product, but we
- -
2
can make it into a -space as follows. For  2 , the temptation is to “absorb”
the scalar into the first coordinate,

²n #³ ~ ² ³ n #
but we must be certain that this is well-defined, that is,
n# ~ n$ ¬ ² ³ n# ~ ² ³n$
But for a fixed , the map ²Á #³ ª ² ³ n # is bilinear and so the universal

property of tensor products implies that there is a unique linear map
n# ª ² ³n#, which we define to be scalar multiplication by .