Sweeping words and the length of a generic vector subspace of M_n(F)

The main result of this short note is a generic version of Paz's conjecture on the lengths of generating sets in matrix algebras. Consider a generic g-tuple A=(A_1,..., A_g) of nxn matrices over a field. We show that whenever $g^{2d}\geq n^2$, the set of all words of degree 2d in A spans the full nxn matrix algebra. Our proofs use generic matrices, are combinatorial and depend on the construction of a special kind of directed multigraphs with few edge-disjoint walks.