I'm reading this paper on infinite matroids and wondering if this theorem from linear algebra also holds for matroids.
Let $E$ be a (possibly infinite) matroid, $I$ be an independent set and $S$ be a spanning set (that is: $cl(S) = E$). Does $|S| \geq |I|$? where $|.|$ denotes cardinality.
The proof of the linear algebra theorem (such as here) uses finiteness in some way, so I don't know how to generalize it to infinite matroids.
My attempt at solving this:
By the $(IM)$ axiom, we can assume that $I$ is a maximal independent set. By lemma 3.10, we have $cl(I) = E$
Since $S$ spans, for all $x \in E$ , either $x \in S$ or there is some independent set $J_x \subseteq S$ such that $J_x + x$ is not independent.
I'm not sure how to connect the $I$ and $J_x$ independent sets together. I think we need to use the $(CL4)$ and $(CLM)$ axioms somehow.