Given that you have $n$ vectors $v$ in some basis (the basis vectors will be denoted with $b$) and you know in what vectors $u$ they are transformed after multiplying them by some matrix A, how can you find the matrix A? The number of vectors ($n$) is equal you the rank of $A$. So:
$Av_i=u_i$ for all $i$ from $1$ to $n$
My idea is to take the matrix $B$ that transforms the basis vectors into the $v$ vectors. (this is trivial):
Let $B$ be a matrix such that $Bb_i=v_i$ for all $i$ from $1$ to $n$
Then we multiply the $u$ vectors by $B$ and we get the $w$ vectors:
$w_i:=B^{-1}u_i$ for all $i$ from $1$ to $n$
If I am not mistaken the $w$ vectors are just the the $u$ vectors but in the basis of the $v$ vectors (so using the $v$ vectors as a basis).
Now we take the matrix $C$ which transforms the basis vectors into the $w$ vectors (this is trivial):
Let $C$ be a matrix such that $Cb_i=w_i$ for all $i$ from $1$ to $n$
I think that A should be calculated in the following way:
$A=BCB^{-1}$
Is this correct?
EDIT: I tried it with an example and it appears it is wrong, but I am not sure why.
EDIT2: I think I found my mistake, so I corrected it. It works with my example now.