He was asked to try 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 (partly because of the rank theorem), and I want to prove it by contradiction. So I'm assuming a $ z in mathcal {R} (B), not in mathcal {R} (A) $, but I can not go beyond this. I am always rounding off the fact that $ z $ It is a linear combination of the rows of $ B $, that both the null space and the row space are based on the stepped form (although the first one is related to the *reduced* stepped form), but that's it. What additional properties should $ z $ So that we can get a contradiction?