# Going from one base packing to another using basis exchanges

Suppose I have a matroid $$M = (E, \mathcal{I})$$. It is a known fact that given any two bases $$X_0$$ and $$X_n$$, we can transform $$X_0$$ into $$X_n$$ by repeatedly applying the basis exchange axiom. So there is a sequence of bases $$X_0, \ldots, X_n$$ such that $$X_{i+1}$$ is obtained by applying basis exchange to $$X_i$$ and $$X_n$$.

I am considering an extension of this. Suppose I have two bases $$A_0$$ and $$B_0$$ which partition the ground set. That is, $$A_0$$ and $$B_0$$ are disjoint bases whose union is the ground set $$E$$.

Now we are given two other bases $$A_n$$ and $$B_n$$. The goal is to transform the partition $$A_0, B_0$$ into $$A_n, B_n$$ by repeatedly applying symmetric basis exchange between $$A_i$$ and $$B_i$$ to obtain $$A_{i+1}$$ and $$B_{i+1}$$. In other words, we are looking for a sequence of partitions $$A_0 \cup B_0, \ldots, A_n \cup B_n$$ such that $$A_i$$ and $$B_i$$ are disjoint bases for each $$i$$ and $$A_i \cup B_i = E$$.

Is this always possible? Do we need to make some assumption about the matroid, such as base orderability?

Thank you