Let $0<r < \min \{m,n\}$ and let $Y_r \subseteq \mathbb{A}^{mn}$ be the determinantal variety of $m$-by-$n$ matrices of rank $\le r$. Let $L \subseteq \mathbb{A}^{mn}$ be a linear variety not containing the origin such that $\dim(Y_r) = \mathrm{codim}(L)$ and $Y_r \cap L$ is nonempty. Is it true that $\dim(L \cap Y_r)=0$ for any such $L$?
Edit after @KReiser’s initial comment: Let us assume that $L$ is not contained in $Y_r$.
(Not sure if this matters, but I only care about the field $\mathbb{C}$. Any references greatly appreciated!)
Final edit: this is indeed still false. Counterexample $r=1$, $L=\begin{pmatrix} c & 1 & 0 \\ d & 1 & 0 \end{pmatrix}$ as $c,d$ vary.