So I was trying this exercise, but I have a lot of doubts:
$P^2\Bbb{(R)}$ projective plane and $x_0,x_1,x_2$ its homogeneous coordinates. Now consider two lines $r: x_0+x_1=0$ and $r': x_0-x_2=0$. Let $\lambda, \mu$ be the homogeneus coordinates of $r$ in the projective frame of $[0,0,1],[1,-1,0]$ and $[1,-1,2]$. Let $\lambda', \mu'$ be the homogeneus coordinates of $r'$ in the projective frame of $[0,1,0],[1,0,1]$ and $[2,3,2]$. Now take $P_0=[1,0,0]$ and $f: r\rightarrow r'$ such that for each $P\in r$, $f$ associate $P'=L(P_0 P)\cap r'$. (where $L(P_0 P)$ is the line from $P_0$ and $P$)
Show $f$ in the coordinates of $\lambda, \mu,\lambda', \mu'$.