  Let $S$ be a set of $n\times n$ matrices over a field $\mathbb{F}$. We show
that the $\mathbb{F}$-linear span of the words in $S$ of length at most
$$2n\log_2n+4n$$ is the full $\mathbb{F}$-algebra generated by $S$. This
improves on the $n^2/3+2/3$ bound by Paz (1984) and an $O\left(n^{1.5}\right)$
bound of Pappacena (1997).
