Prove that if we work with the field $\mathbb{C}$, an homography defined in a line $l$ of the projective plane $h : l\to l$ can be expressed as a product of two perspectivities. With $\mathbb{K}=\mathbb{R}$, give and example of a homography that required to be the result of the product of three perspectivities.
I studied that an homography $h:l \to l$ can be expressed as a product of three perspectivities and I understood the proof, but I don't see what changes in $\mathbb{C}$.
In $\mathbb{C}$ every projectivity has a fixed point. Let $F\in l$ be a fixed point of $h$. Let $s$ be an arbitrary line through $F$ and $P$ be an arbitrary point outside $l$ and $s$. If $A$ and $B$ are two points on $l$ let $A'$ be $PA\cap s$, $B'$ be $PB\cap s$ and $Q=A'h(A)\cap B'h(B)$. Then the product of projectivities $$l\overset{P}{\doublebarwedge}s\overset{Q}{\doublebarwedge}l$$ is $h$.