Let $M_1=(E,\mathcal{I}_1)$ and $M_2=(E,\mathcal{I}_2)$ be two matroids defined on the same ground set $E$ with independent sets $\mathcal{I}_1$ and $\mathcal{I}_2$ respectively. Suppose we can check whether a subset $S$ of $E$ is an independent set of $M_1$ and $M_2$ in unit time. How can we efficiently find a base of matroid $M_1$ which is not a base of $M_2$?
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