  We consider the definability of mad families in vector spaces of the form
$\underset{n<\omega}{\bigoplus} F$ where $F$ is a field of cardinality $\leq
\aleph_0$. We show that there is no analytic mad family of subspaces when
$F=\mathbb{F}_2$, partially answering a question of Smythe. Our proof relies on
a variant of Mathias forcing restricted to a certain idempotent ultrafilter
whose existence follows from Glazer's proof of Hindman's theorem.
