Lines of full rank matrices in large subspaces

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.

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