On the matrices of given rank in a large subspace

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.

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