\b[False.] Triviality of $\mathcal N(A)$ gives $r=n$, while triviality of $\mathcal N(A^T)$ gives $r=m$ because <transpose> preserves <matrix rank>. Both conditions therefore force $m=n$, contradicting the stated unequal dimensions.

Solution

Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof resit / 5 / a / iv / Solution

 Solution (source code)