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Linear independence preservation

Source: IberoAmerican Olympiad For University Students

July 10, 2011
vectorlinear algebramatrixlinear algebra unsolved

Problem Statement

For each pair of integers (i,k)(i,k) such that 1≤i≤k1\le i\le k, the linear transformation Pi,k:Rk→RkP_{i,k}:\mathbb{R}^k\to\mathbb{R}^k is defined as:
Pi,k(a1,⋯ ,ai−1,ai,ai+1,⋯ ,ak)=(a1,⋯ ,ai−1,0,ai+1,⋯ ,ak)P_{i,k}(a_1,\cdots,a_{i-1},a_i,a_{i+1},\cdots,a_k)=(a_1,\cdots,a_{i-1},0,a_{i+1},\cdots,a_k)
Prove that for all n≥2n\ge2 and for every set of n−1n-1 linearly independent vectors v1,⋯ ,vn−1v_1,\cdots,v_{n-1} in Rn\mathbb{R}^n, there is an integer kk such that 1≤k≤n1\le k\le n and such that the vectors Pk,n(v1),⋯ ,Pk,n(vn−1)P_{k,n}(v_1),\cdots,P_{k,n}(v_{n-1}) are linearly independent.
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