The rank of the incidence matrix of the lattice of subsets of a finite set or the lattice of subspaces of a finite dimensional vector space over the finite field \(F_ q\), \(q=p^ d\), is computed over any field \(K\), except when \(\hbox{char}(K)=2\) in the case of subsets and \(\hbox{char}(K)=p\) in the case of subspaces. The proof of the result is based on some facts from the representation theory of the finite symmetric and general linear groups as developed in the books of G. James. As may be expected the rank formula for subsets is obtained from the corresponding formula for subspaces by taking \(q=1\).