Action of $PGL(n+1) $ on $\mathbb P^n$!

Question

what is the action of $PGL(n+1)$ on Projective space $\mathbb P^n$? (over algebraic closed field)

Answer 1

$\mbox{GL}(n+1)$ acts on $k^{n+1}\backslash\{0\}$ by matrix-vector multiplication. By taking quotient by the group $k^*$, this makes the action of $\mbox{GL}(n+1)$ on $k^{n+1}\backslash\{0\}$ descend to an action of $\mbox{PGL}(n+1)$ on $\mathbb{P}^n$. Explicitly, an element $[A]\in\mbox{PGL}(n+1)$ induced by a matrix $A$ acts on $[x_0:\cdots:x_n]$ by $$[A]\cdot[x_0:\cdots:x_n]=\pi\left(A\left(\begin{array}{c}x_0\\\vdots\\x_n\end{array}\right)\right),$$ where $\pi:k^{n+1}\backslash\{0\}\to\mathbb{P}^n$ is the natural projection.