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In 2D, a linear transformation has the form $v \mapsto Mv$ where $M$ is a $2\times 2$ matrix, so the transformation has four parameters. Thus, the transformation is uniquely determined if you have two points $p_1,p_2$ from the first set and two points $q_1,q_2$ and if you know that $p_1$ transforms to $q_1$ and $p_2$ to $q_2$. You can find this linear ...

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