If you have an independent set of vectors $\left\{v_{i_1},\ldots , v_{i_{n-1}} \right\}$, what does it mean for a vector to be "supported on the complement of $\left\{i_1, \ldots , i_{n-1} \right\}$"?
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Maybe this one is a duplicate, or nearly the same: http://math.stackexchange.com/questions/489517/support-of-a-vector – Attila Jan 24 '14 at 15:49