Consider two matrices $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{k\times n}$ such that $$ \text{rank}\left[\begin{array}{c} A \\ B \end{array}\right] = \text{rank}\, A $$ which is to say that the span of the rows of $B$ is contained in the span of the rows of $A$. That is, we can write this condition in terms of the range of matrices as $\mathcal{R}(A^T)\supseteq\mathcal{R}(B^T)$.
Question: Is it then also equivalent to write the condition in terms of the nullspace of matrices as $\mathcal{N}(B)\supseteq\mathcal{N}(A)$?
I think this is true because $\mathcal{R}(A^T) = \mathcal{N}(A)^\perp$. But I am not fully convinced, I would like a proof or disproof, or any reference that you can provide. I will be very thankful.
If $U,V \leq \mathbb{R}^l$ are subspaces then $U \subseteq V$ iff $U^{\perp} \supseteq V^{\perp}$. Together with your observation, we have
$$ \mathcal{R}(B^T) \subseteq \mathcal{R}(A^T) \iff \mathcal{N}(B)^{\perp} \subseteq \mathcal{N}(A)^{\perp} \iff \mathcal{N}(B) \supseteq \mathcal{N}(A). $$