Matroids with given restrictions and contractions

If M is a matroid on a set S and if X is a subset of S, then there are two matroids on X induced by M: namely, the restriction and the contraction of M onto X. Necessary and sufficient conditions are obtained for two matroids on the same set to be of this form and an analogous result is obtained when (X₁,…, X_t) is a partition of S. The corresponding results when all the matroids are binary are also obtained.