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Random Permutation Matrices Form a Basis with High Probability
Let $d_n=(n-1)^2+1$, the dimension of the real linear span of the $n\times n$ permutation matrices. We prove that $d_n$ independent uniformly random permutation matrices are linearly independent with probability $1-O(n^{-1/2})$. Conditioning on distinctness gives the same conclusion for a uniformly
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