Given two sets of dimension $n$ vectors
$\lbrace v_1 , v_2 , \ldots , v_m \rbrace$, $\lbrace u_1, u_2, \ldots , u_m \rbrace$,
where $m > n$, is there a computational method (in particular, using a program such as Mathematica, Maple, etc) to find an $n \times n$ matrix $A$ that gives a bijection between the two sets, so $A v_i = u_j$ for some $i,j$. In particular, the $n \times n$ matrix must have determinant $\pm 1$. The span of each set is the full $n$-dimensional space.
I have two sets of vectors that I think are the same up to a change in basis that preserves volume and permutes the vectors, but I can't think of an algorithmic way to approach the question.