It was asked to prove the following statement:

$ \mathcal{N}(A) \subseteq \mathcal{N}(B) \rightarrow \mathcal{R}(B^\intercal) \subseteq \mathcal{R}(A^\intercal)$

For me, it is intuitively true (somewhat because of the rank theorem), and I want to prove by contradiction. So I am supposing a $ z \in \mathcal{R}(B), \not\in \mathcal{R}(A)$ , but I can’t go on beyond this. I am always rounding the fact that $ z$ is a linear combination of the rows of $ B$ , that the null space and the row space are both founded in the echelon form (though the former is related to the *reduced* echelon form), but that is it. What additionals properties do $ z$ must have so we can get a contradiction?