Given two different bases $\{v_1,..., v_n\}$ and $\{w_1,...,w_n\}$ of a vector space $V$, show that for any $v_i$, exists $w_j$ such that both $(\{v_1,..., v_n\}\backslash\{v_i\})\cup\{w_j\}$ and $(\{w_1,..., w_n\}\backslash\{w_j\})\cup\{v_i\}$ are also basis.

I understand that in the context of matroid, this is known as the strong exchange property. However, I haven't learned anything about matroid yet and need to prove this directly using linear algebra.

To show half of the result is easy using the basic result of exchange lemma. However, it is tricky to show that the found vector $w_j$ simultaneously satisfies both sets to be bases, which is where I am stuck.


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