Let {a_{1},a_{2},...,a_{n}} and {b_{1},b_{2},...,b_{n}} be two bases for a vector space E. Fix p, 1 ≤ p ≤n. Is there a permutation σ such that
{a_{1},a_{2},...,a_{p},b_{σ(p+1)},...,b_{σ(n)}} and {b_{σ(1)},b_{σ(2)},...,,b_{σ(p)},a_{p+1},...,a_{n}} are both bases of E?

This question is the last exercise of the first chapter in the book Linear Algebra by Greub. I can prove the case p=n-1.