[math-fun] Find the matrix

Given two (unordered) sets of (binary) integer vectors. Considerations of symmetry lead to the conjecture that there exists a linear bijection (indeed, a permutation of the basis) relating them. Design an (efficient) algorithm to construct some such matrix. In the case to hand, the dimension equals 20 (reducible to nearer 10 after some finagling), and vector sets have size 120 and full rank. Fred Lunnon