Page 390 - Advanced Linear Algebra

374 Advanced Linear Algebra

'~ n ! n "

0
has rank .

Defining Linear Transformations on a Tensor Product
One of the simplest and most useful ways to define a linear transformation on

the tensor product <n = is through the universal property, for this property
says precisely that a bilinear function on < d = gives rise to a unique (and

well-defined) linear transformation on <n = . The proof of the following
theorem illustrates this well.

Theorem 14.7 Let < and = be vector spaces. There is a unique linear
transformation
i
i
¢< n = ¦ ²< n = ³ i
defined by ² n ³ ~ p where
² p ³²" n#³ ~ ²"³ ²#³
Moreover, is an embedding and is an isomorphism if and are finite-

<
=
(
dimensional. Thus, the tensor product n of linear functionals is via this
embedding a linear functional on tensor products.
)

Proof. Informally, for fixed and , the function " ² Á # ³ ¦ ² " ³ ² # ³ is bilinear
in and and so there is a unique linear map p taking n " # to ² " ³ ² # . ³
#
"

The function ² Á ³ ¦ p is bilinear in and since
² b ³ p ~ ² p ³ b ² p ³

and so there is a unique linear map taking n to p .

-
More formally, for fixed and , the map Á ¢ < d = ¦ - defined by
- Á ²"Á #³ ~ ²"³ ²#³
is bilinear and so the universal property of tensor products implies that there
exists a unique p  ²< n = ³ i for which
² p ³²" n#³ ~ ²"³ ²#³
i
i
Next, the map .¢ < d = ¦ ²< n = ³ i defined by
.² Á ³ ~ p