Suppose that $S_1, \ldots, S_n$ are finite subsets of $\mathbb{R}^n$. Suppose that for every subset $J= \{j_1, \ldots, j_\ell\} \subset \{1, \ldots, n\},$ the dimension of $span(S_{j1} \cup S_{j2} \cup \cdots \cup S_{j\ell})$ is at least $\ell$. Show that one can choose vectors $v_1 \in S_1, v_2 \in S_2, \ldots, v_n \in S_n$ such that $\{v_1, \ldots, v_n\}$ is a basis for $\mathbb{R}^n$.
The information about the span of the union of subsets sounds like it could make Hall's Marriage theorem useful. Hall can be deduced from Dilworth, so I was considering that too, for example maybe creating a poset related to whether a vector $v$ is in the span of another set of vectors. Vector spaces have more structure than the problems I've solved with Hall before, so I'm finding it hard to find the right way to proceed. I would appreciate a bump in the right direction.