  Let V be a linear subspace of M_{n,p}(K) with codimension lesser than n,
where K is an arbitrary field and n >=p. In a recent work of the author, it was
proven that V is always spanned by its rank p matrices unless n=p=2 and K is
isomorphic to F_2. Here, we give a sufficient condition on codim V for V to be
spanned by its rank r matrices for a given r between 1 and p-1. This involves a
generalization of the Gerstenhaber theorem on linear subspaces of nilpotent
matrices.
