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.



On permutation of elements of two bases of a vector space (Greub´s book)
It's also Theorem 7.2 in [Prasolov's Problems in Linear Algebra](http://www2.math.su.se/~mleites/books/prasolov-1994-problems.pdf), which gives a proof and attributes it to Green 1973.


