The classification of large spaces of matrices with bounded rank

Given an arbitrary (commutative) field K, let V be a linear subspace of M_n(K) consisting of matrices of rank lesser than or equal to some r<n. A theorem of Atkinson and Lloyd states that, if dim V>nr-r+1 and #K>r, then either all the matrices of V vanish on some common (n-r)-dimensional subspace of K^n, or it is true of the matrices of its transpose V^t. Following some arguments of our recent proof of the Flanders theorem for an arbitrary field, we show that this result holds for any field. We also show that the results of Atkinson and Lloyd on the case dim V=n\,r-r+1 are independent on the given field save for the special case n=3, r=2 and #K=2, where the same techniques help us classify all the exceptional cases up to equivalence. Similar theorems of Beasley for rectangular matrices are also successfully extended to an arbitrary field.