TITLE: Is a vector bundle orientable if and only if its dual bundle is orientable?
QUESTION [7 upvotes]: 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)?

REPLY [4 votes]: Alternatively, we can make two observations:

A bundle $E$ of dimension $n$ is orientable iff $\Lambda^n E$ is trivial. 
If we denote $F^*$ the bundle dual to a vctor bundle $F$, then there is a isomorphism $\Lambda^n(E^*)\to(\Lambda^nE)^*$.