Given a set of point $P$ in $\mathbb R^n$ and the same set of points $P'$ which have been transformed by a transformation matrix: $$L: \mathbb R^n\mapsto \mathbb R^n$$ $$L(p_1) = p_1',\;\; p_1\in P\land p_1 '\in P'$$ What is the minimum amount of point pairs ($p_1,p_1'$) required to find $L$ for a given $n$? And what are the requirements for these points?
I think they should not lie on one line or plane (depending on $n$)
You need $n$ linearly independent points in $\mathbb R^n$ to determine $L$. If you know what $L$ does to a set $P$ of points in $\mathbb R^n$ then linearity tells you what $L$ does to every linear combination of points in $P$. The space $\mathbb R^n$ has sets of $n$ linearly independent points but not any sets of more than $n$ points. If you pick $k<n$ pairs of points then there will be sets of at least $n-k$ points in $\mathbb R^n$ for which $L$ is not determined and we won't know what the transformation does. If you pick $k>n$ pairs of points, then some will be linearly dependent and thus either redundant or contradictory.