  In this short note we prove a lemma about the dimension of certain algebraic
sets of matrices. This result is needed in our paper arXiv:1201.1672. The
result presented here has also applications in other situations and so it
should appear as part of a larger work.
  The statement of the lemma goes as follows: Suppose $X$ is a nonempty
algebraically closed subset of the affine space of $n \times m$ complex
matrices. Suppose that $X$ is column-invariant (i.e., belongingness to $X$
depends only on the column space of the matrix). Suppose $E$ is a vector
subspace of $\mathbb{C}^n$ that is not contained in the column space of any
matrix in $X$. Then $codim(X) \geq m + 1 - dim(E)$.
  The proof is simple and relies on intersection theory of the grassmannians
("Schubert calculus").
