Let $f:G(k, n)\to G(n-k, n)$ be the map taking a $k$-plane $V$ (i.e., a $k$-dimensional subspace of $\mathbb{R}^n$) into it's orthogonal complement $V^{\perp}$. Show that $f$ is a diffeomorphism.
My question is whether or not the following strategy can work:
Since $f$ is obviously a bijection, it is enough to prove that $f$ is a local diffeomorphism, so the plan is to prove that the linear map $f_{*_{p}}$ has full rank $k(n-k)$ for every $p\in G(k, n)$.
My difficulty is: I'm having trouble to calculate $f_{*_{p}}(v)$ for an arbitrary $v \in T_p(G(k, n))$ because I don't know how to concretely represent such a $v$.
Is this strategy doable? Thanks!