Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics I / 2 / a / ii

Suppose ${\mathbf{x}}_0$ is one solution. By linearity,Thus the solution set of a linear system is exactly ${\mathbf{x}}_0+\ker T_A$. It has one member if the kernel is $\{0\}$. Conversely, if there is a unique solution and $\mathbf{z}\in \ker T_A$, then ${\mathbf{x}}_0+\mathbf{z}$ is also a solution, forcing $\mathbf{z}=0$. Combining this with i gives