Maximal rank in matrix spaces via graph matchings

It is shown that if S is a translate of a linear space contained in Wn(F) and |F|≥3 then ρ(S)≥μ(GS). The restriction on F can be removed if S is an affine subspace of An(F). Applications include simple proofs of upper bounds on the dimension of affine subspaces of symmetric and alternating matrices of bounded rank.