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

If $\mathbf{b}$ is outside the image of a linear map $T_A$, there is no solution. Otherwise choose ${\mathbf{x}}_0$ with $A{\mathbf{x}}_0=\mathbf{b}$. By ii, a zero kernel gives a unique solution. If the kernel contains a nonzero vector $\mathbf{z}$, allare solutions. They are distinct, since $(t-s)\mathbf{z}=0$ forces $t=s$. Hence there are infinitely many. These alternatives exhaust all possibilities.