Subspaces of matrices with special rank properties

Let K be a field and let Mm×n(K) denote the space of m×n matrices over K. We investigate properties of a subspace M of Mm×n(K) of dimension n(m-r+1) in which each non-zero element of M has rank at least r and enumerate the number of elements of a given rank in M when K is finite. We also provide an upper bound for the dimension of a constant rank r subspace of Mm×n(K) when K is finite and give non-trivial examples to show that our bound is optimal in some cases. We include a similar a bound for the maximum dimension of a constant rank subspace of skew-symmetric matrices over a finite field.