  Suppose that $f$ is a homomorphism from the mapping class group
$\mathcal{M}(N_{g,n})$ of a nonorientable surface of genus $g$ with $n$
boundary components, to $\mathrm{GL}(m,\mathbb{C})$. We prove that if $g\ge 5$,
$n\le 1$ and $m\le g-2$, then $f$ factors through the abelianization of
$\mathcal{M}(N_{g,n})$, which is $\mathbb{Z}_2\times\mathbb{Z}_2$ for
$g\in\{5,6\}$ and $\mathbb{Z}_2$ for $g\ge 7$. If $g\ge 7$, $n=0$ and $m=g-1$,
then either $f$ has finite image (of order at most two if $g\ne 8$), or it is
conjugate to one of four "homological representations". As an application we
prove that for $g\ge 5$ and $h<g$, every homomorphism
$\mathcal{M}(N_{g,0})\to\mathcal{M}(N_{h,0})$ factors through the
abelianization of $\mathcal{M}(N_{g,0})$.
