I was reading up on my dual spaces today and I made the following hypothesis:
A vector bundle $\xi$ is orientable if and only if $\xi^*$ is orientable.
This seems rather intuitive, and although it doesn't seem too hard to prove, I'm not sure how to formally prove it. Any help?
EDIT: Also, I guess another assertion that would fall along these lines would be whether the following is true:
If the ordered bases $v_1,...,v_n$ and v'_1,...,v'_n for $V$ are equally oriented, then the same is true of the bases $v^*_1,...,v^*_n$ and v^{'*}_1,...,v^{'*}_n for $V^*$.
Can anyone help me on the proofs (if they're true)?