Let $P,U$ be points in $K^2$ ($K$ is a field). Let $L(1)$, $L(2)$ be two lines through $P$, and $L(3)$, $L(4)$ be two distinct lines through $U$.
How to show that there is an affine change of coordinates $T$ of $K^2$ such that $T(P)=U$, $T(L(1))=L(3)$, and $T(L(2))=L(4)$?