Let

be a bilinear map, where

and

are vector spaces over a field

. Recall that the dual vector space is denoted by

. Let

be subspaces of V.

i). Prove that

Prove what? You only wrote the definition of orthogonal complement. Perhaps it was to prove that this complement is a subspace?
My answer:

.

Hence

by the definition of

.

We also have that

from the definition of

.

Hence we have that

so

is a subspace of W.

ii). Prove that

.

This is where things start to go a bit wrong. I have no idea what

is defined to be!

== ==
iii). Prove that

I'm also a little stuck on this as well. I decided to start by letting

.

I also know that

.

However, I wasn't sure where this would take me =S

, and this is true is particular for all since !
iv). Given that

defined by

for

is a linear map from

to

, and that

, deduce that

I figure this has something to do with rank+ nullity....

Of course! And it also has to do with the important fact that for finite dimensional spaces, so by the rank+nullity theorem: Tonio
I'd really appreciate any help with this. I'm finding the concept of bilinear maps and dual vector spaces quite hard =S