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

True. The rank-nullity theorem gives $\dim \mathcal{N}(A)=n-r=0$. The only vector in this zero-dimensional kernel of a linear map is the zero vector.

False. Take $A=(1,0{)}^T$, an $m=2$, $n=1$ matrix of matrix rank one, and $b=(0,1{)}^T$. Then $Ax=(x,0{)}^T$ never equals $b$. Full column rank implies injectivity, not surjectivity onto a larger codomain.

True. A square matrix with matrix rank $n$ is invertible, and its transpose has the same rank. Applying the rank-nullity theorem to both $A$ and $A^T$ gives nullity zero in each case, so both kernels are $\{0\}$.

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.