Question

Let {a₁,a₂,...,a_n} and {b₁,b₂,...,b_n} be two bases for a vector space E. Fix p, 1 ≤ p ≤n. Is there a permutation σ such that {a₁,a₂,...,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.

Answer 1

It's also Theorem 7.2 in Prasolov's Problems in Linear Algebra, which gives a proof and attributes it to Green 1973.