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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 $T(L(2))=L(4)$?

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Hint: You have only one choice for the coordinate change, all you need is show that it is affine linear.

You can write $L(1)=P+tv_1$ for some $v_1\in k^2$ and similarly for $L(2), L(3), L(4)$. What does the fact that $L(1)$ and $L(2)$ are distinct tell you about $v_1$ and $v_2$, similarly for $v_3$ and $v_4$? Where do you map $P$, where $v_1$ and $v_2$?

Can you realise this as a matrix (+translation)?