On the definability of mad families of vector spaces

We consider the definability of mad families in vector spaces form ⨁ n < ω F where is a field cardinality ≤ ℵ 0 . show that there no analytic family subspaces when = 2 , partially answering question Smythe. Our proof relies on variant Mathias forcing restricted to certain idempotent ultrafilter whose existence follows from Glazer's Hindman's theorem.