  Recently, John Franks and Michael Handel proved that, for $g\geq 3$ and
$n\leq 2g-4$, every homomorphism from the mapping class group of an orientable
surface of genus $g$ to $\GL (n,\C)$ is trivial. We extend this result to
$n\leq 2g-1$, also covering the case $g=2$. As an application, we prove the
corresponding result for nonorientable surfaces. Another application is on the
triviality of homomorphisms from the mapping class group of a closed surface of
genus $g$ to $\Aut (F_n)$ or to $\Out (F_n)$ for $n\leq 2g-1$.
