We consider a finite-dimensional vector space $W\subset K^E$ over an arbitrary field $K$ and an arbitrary set $E$. We show that the set $C(W)\subset 2^E$ consisting of the minimal supports of $W$ are the circuits of a matroid on $E$. In particular, we show that this matroid is cofinitary (hence, tame). When the cardinality of $K$ is large enough (with respect to the cardinality of $E$), then the set $trop(W)\subset 2^E$ consisting of all the supports of $W$ is a matroid itself. Afterwards we apply these results to tropical differential algebraic geometry and study the set of supports $trop(Sol(\Sigma))\subset (2^{\mathbb{N}^{m}})^n$ of spaces of formal power series solutions $\text{Sol}(\Sigma)$ of systems of linear differential equations $\Sigma$ in differential variables $x_1,\ldots,x_n$ having coefficients in the ring ${K}[\![t_1,\ldots,t_m]\!]$. If $\Sigma$ is of differential type zero, then the set $C(Sol(\Sigma))\subset (2^{\mathbb{N}^{m}})^n$ of minimal supports defines a matroid on $E=\mathbb{N}^{mn}$, and if the cardinality of $K$ is large enough, then the set of supports $trop(Sol(\Sigma))$ itself is a matroid on $E$ as well. By applying the fundamental theorem of tropical differential algebraic geometry (fttdag), we give a necessary condition under which the set of solutions $Sol(U)$ of a system $U$ of tropical linear differential equations to be a matroid. We also give a counterexample to the fttdag for systems $\Sigma$ of linear differential equations over countable fields. In this case, the set $trop(Sol(\Sigma))$ may not form a matroid.

翻译：考虑一个任意域$K$上的有限维向量空间$W\subset K^E$，其中$E$为任意集合。我们证明，由$W$的最小支撑集构成的集合$C(W)\subset 2^E$是$E$上拟阵的圈集。特别地，我们证明该拟阵是余有限的（因而也是驯顺的）。当域$K$的基数足够大（相对于$E$的基数）时，由$W$的所有支撑集构成的集合$trop(W)\subset 2^E$本身构成一个拟阵。随后，我们将这些结果应用于热带微分代数几何，并研究具有系数在环${K}[\![t_1,\ldots,t_m]\!]$中的线性微分方程组$\Sigma$（微分变量为$x_1,\ldots,x_n$）的形式幂级数解空间$\text{Sol}(\Sigma)$的支撑集集合$trop(Sol(\Sigma))\subset (2^{\mathbb{N}^{m}})^n$。若$\Sigma$的微分类型为零，则由最小支撑集构成的集合$C(Sol(\Sigma))\subset (2^{\mathbb{N}^{m}})^n$在$E=\mathbb{N}^{mn}$上定义了一个拟阵；且当域$K$的基数足够大时，支撑集集合$trop(Sol(\Sigma))$本身也是$E$上的一个拟阵。通过应用热带微分代数几何基本定理，我们给出了热带线性微分方程组$U$的解集$Sol(U)$构成拟阵的一个必要条件。同时，我们给出了该基本定理在可数域上线性微分方程组$\Sigma$情形下的反例：此时$trop(Sol(\Sigma))$可能不构成拟阵。