The e-exchange basis graph and matroid connectedness

Let $M$ be a matroid and $e\in E\left(M\right)$. The $e$-exchange basis graph of $M$ has vertices labeled by bases of $M$, and two vertices are adjacent when the bases labeling them have symmetric difference $\{e,x\}$ for some $x\in E\left(M\right)$. In this paper we show that a connected matroid is exactly a matroid with the property that for every element $e\in E\left(M\right)$, the $e$-exchange basis graph is connected.