Grassmanians and Plucker embedding

Question

Are the Grassmannians $G(r,n)$ and $G(n-r,n)$ isomorphic as projective varieties ? I can see that they have the same dimension $r(n-r)$. If they are isomorphic then I would like to see them isomorphic as images of their respective Plucker embedding.

Answer 1

Yes they are. One way to see this is to note that every $r$ plane $W$ in $V \cong k^n$ gives an $n -r $ plane in $V^* \cong k^n$, by sending $W$ to the space of functionals vanishing on it. You can probably check in local affine coordinates on the Grassmannian (the patches look like $Hom(k^r, k^{n -r})$) that this map is algebraic. This can be pictured as the orthogonal complement to a subspace.