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Let {a1,a2,...,an} and {b1,b2,...,bn} be two bases for a vector space E. Fix p, 1 ≤ p ≤n. Is there a permutation σ such that {a1,a2,...,ap,bσ(p+1),...,bσ(n)} and {bσ(1),bσ(2),...,,bσ(p),ap+1,...,an} 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.

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COuld you explain why such a fact is interesting/useful? – Mariano Suárez-Alvarez Jan 8 '10 at 1:31
No. I´ve no idea why anyone would care about it. It´s just an exercise in a book of linear algebra. – Julio Cesar da Silva Jan 8 '10 at 14:59
up vote 4 down vote accepted

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

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Included link is broken. – Turbo Apr 10 '15 at 5:06

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