.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:
I'd really appreciate any help with this. I'm finding the concept of bilinear maps and dual vector spaces quite hard =S