  Let $n$ and $p$ be non-negative integers with $n \geq p$, and $S$ be a linear
subspace of the space of all $n$ by $p$ matrices with entries in a field
$\mathbb{K}$. A classical theorem of Flanders states that $S$ contains a matrix
with rank $p$ whenever $\mathrm{codim} S <n$.
  In this article, we prove the following related result: if $\mathrm{codim}
S<n-1$, then, for any non-zero $n$ by $p$ matrix $N$ with rank less than $p$,
there exists a line that is directed by $N$, has a common point with $S$ and
contains only rank $p$ matrices.
