$SU(2)$ acts transitively on $\mathbb{CP}^1$

Question

How can one proof rigorously that the action of $SU(2)$ on $\mathbb{CP}^1$, where $\mathbb{CP}^1$ is the complex projective space, is transitive? i.e., that for any $u, v \in \mathbb{CP}^1$, there exists an $A \in SU(2)$ such that $A(u)=v$.

I can't seem to formulate a proof; other than that the statement is obvious?

Answer 1

This follows from the fact that, given any two vectors $v$ and $w$ in $\mathbb{C}^2$ with $\|v\|=\|w\|=1$, there is some $g\in SU(2)$ such that $g.u=v$.