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Let σ be the vector of ones. Let V be the set of vectors whose largest entry is 1 and whose smallest entry is 0.

When is it (generically) the case that the set of vectors {σ, v_1, v_2, ..., v_n} is linearly independent, where v_i are vectors randomly drawn from V?

I am guessing that this is so whenever m > n + 1, but am not sure how to prove it.

Any help is much appreciated, thank you!

JF