Page 376 - Advanced Linear Algebra

360 Advanced Linear Algebra

is bilinear if it is linear in both variables separately, that is, if
Z
Z
² " b " Á #³ ~ ²"Á #³ b ²" Á #³
and
Z
Z
²"Á # b # ³ ~ ²"Á #³ b ²"Á # ³
The set of all bilinear functions from <d = to > is denoted by
hom - ²<Á= Â> ³. A bilinear function ¢< d = ¦ - with values in the base
field is called a bilinear form on < d = .
-
Note that bilinearity can also be expressed in matrix language as follows: If

~ ² ÁÃÁ ³  - Á ~ ² ÁÃÁ ³  -

and

" ~ ²" ÁÃÁ" ³  < Á # ~ ²# ÁÃÁ# ³  =

then ¢ < d = ¦ > is bilinear if
!
!
² " Á # ³ ~ - !
where - ~ ´ ²" Á # ³µ Á .

It is important to emphasize that, in the definition of bilinear function, <d = is
the cartesian product of sets , not the direct product of vector spaces. In other
words, we do not consider any algebraic structure on <d = when defining
bilinear functions, so expressions like
²%Á &³ b ²'Á $³ and ²%Á &³
are meaningless.

In fact, if is a vector space, there are two classes of functions from = d = to
=
> : the linear maps B ²=d = Á > ³, where =d = ~ = ^ = is the direct
product of vector spaces, and the bilinear maps hom²= Á= Â> ³ , where = d = is
just the cartesian product of sets. We leave it as an exercise to show that these
two classes have only the zero map in common. In other words, the only map
that is both linear and bilinear is the zero map.

We made a thorough study of bilinear forms on a finite-dimensional vector
)
(
space in Chapter 11 although this material is not assumed here . However,
=
bilinearity is far more important and far-reaching than its application to metric
vector spaces, as the following examples show. Indeed, both multiplication and
evaluation are bilinear.
Example 14.4 (Multiplication is bilinear ) If is an algebra, the product map
(
¢( d ( ¦ ( defined by