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

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.