On the definability of mad families of vector spaces

We consider the definability of mad families in vector spaces of the form ⨁n<ωF where F is a field of cardinality ≤ℵ0. We show that there is no analytic mad family of subspaces when F=F2, 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.