Let $V$ be a finite-dimensional left vector space over a division ring $K$, and let $V^*$ the dual right vector space (consisting of all linear functions from $V$ to $K$). We can (and will) treat $V^*$ as a left vector space over the division ring $K^*$ (obtained from $K$ by reversing the arguments of the multiplication).
If $K$ is a field, then we have $V \cong V^*$. If $K$ is only a division ring, we can of course not have this as $V^*$ is not a left vector space over $K$ but over $K^*$.
However: Does ist hold that $P(V)$ and $P(V^*)$ are still isomorphic? That is, are the projective spaces consisting of the one- and two-dimensional vector spaces still isomorphic? And if so, could you please give me the collineation and explain why it works?