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' conjecture on the length of generating sets in matrix algebras. Consider a generic g-tuple A _ = ( A 1 , ź , A g ) of n × n matrices over an infinite field. We show that whenever g 2 d ź n 2 , the set of all words of degree 2d in A _ spans the full n × n matrix algebra. Our proofs use generic matrices, are combinatorial and depend on the construction of special kinds of directed multigraphs with few edge-disjoint walks.