Is every collineation of the real projective plane given by some linear transformations?

Question

Suppose we have some collineation of the real projective plane $\alpha$. Is it possible that $\alpha$ does not use linear transformations? My thoughts are that it isn't possible. This is because we can just add lines and planes normally (points and lines). No matter the points P and Q. We'll always have $\alpha$(P + Q) = $\alpha$(P) + $\alpha$(Q). Similarly, the same thing can be said about lines. We also know that for some constant $\lambda$ which isn't 0 we have the $\lambda$P = P because it's just a constant multiple of some vector (point) P. So constants pass through $\alpha$ as well. Is there something I'm missing? Can anyone give me an example of a collineation that isn't a linear transformation?

Answer 1

Yes. According the Fundamental theorem of projective geometry, collineations of a projective space of a vector space $V$ over a field $K$ having dimension greater than $1$ are generated by linear transformations of $V$ and field automorphisms of $K$. Since the field of real numbers ($\mathbb R$) doesn't have any nontrivial field automorphism, collineations of a projective space over a real vector space are exactly the projective linear transformations (i.e. the transformations generated by the linear transformations of $V$). But e.g. in the case when $K=\mathbb C$ (the field of complex numbers), an example for a nontrivial field automorphism is the complex conjugation. So, an example of a collineation that isn't a projective linear transformation is the transformation generated by the complex conjugation of the coordinates of the vectors of $V$.