Vandermonde Matrices, NP-Completeness, and Transversal Subspaces

Let K be an infinite field. We give polynomial time constructions of families of r-dimensional subspaces of Kn with the following transversality property: any linear subspace of Kn of dimension n− r is transversal to at least one element of the family. We also give a new NP-completeness proof for the following problem: given two integers n and m with n m and a n ×m matrix A with entries in Z, decide whether there exists a n× n sub-determinant of A which is equal to zero.