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Nullspace Transpose[{{1,1,1,1,1,1,1,1},{1,1,-1,-1,1,1,1,1},{1,1,-1,1,-1,1,-1,-1},{1,1,1,-1,-1,1,-1,-1},{2,-1,0,0,2,-1,-1,-1},{2,-1,0,0,-2,-1,1,1},{-4,4,0,0,0,-4,0,0},{-8,-4,0,0,0,4,0,0}}]
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