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 K-n with the following transversality property: any linear subspace of K-n 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 less than or equal to m and a n x m matrix A with entries in Z, decide whether there exists an n x n subdeterminant of A which is equal to zero.