We consider a finite-dimensional vector space $W\subset K^E$ over a field K and a set E. We show that the set $\mathcal {C}(W)\subset 2^E$ of minimal supports of W are the circuits of a matroid on E. When the cardinality of K is large (compared to that of E), then the family of supports of W is a matroid. Afterwards we apply these results to tropical differential algebraic geometry (tdag), studying the set of supports of spaces of formal power series solutions $\text {Sol}(\Sigma )$ of systems of linear differential equations (ldes) $\Sigma$ in variables $x_1,\ldots ,x_n$ having coefficients in . If $\Sigma $ is of differential type zero, then the set $\mathcal {C}(Sol(\Sigma ))\subset (2^{\mathbb {N}^{m}})^n$ of minimal supports defines a matroid on $E=[n]\times \mathbb {N}^{m}$, and if the cardinality of K is large enough, then the set of supports is also a matroid on E. By applying the fundamental theorem of tdag (fttdag), we give a necessary condition under which the set of solutions $Sol(U)$ of a system U of tropical ldes is a matroid. We give a counterexample to the fttdag for systems $\Sigma $ of ldes over countable fields for which is not a matroid.