Linear transformations on matrices: the invariance of sets of ranks

Let R_E denote the set of all m × n matrices over an algebraically closed field F whose ranks lie in the set E, where E is a subset of {1,2,…,m}. Let T be a linear transformation which maps R_E into itself. Under some restrictions on E, or when T is nonsingular, there are nonsingular matrices U and V such that T(A) = UAV for every m × n matrix A.