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

No. Here $A$ has more rows than columns, which does not force a nonzero kernel. For example, with $m=1$, $n=2$,the linear system has the unique solution $x=0$. With the same $A$ and $\mathbf{b}=(0,1{)}^T$ it has no solution at all.

No. Take $A=I_m$ and $\mathbf{b}=0$. The homogeneous linear system then has only the solution $\mathbf{x}=0$. Homogeneity ensures existence, but infinitely many solutions would additionally require a nonzero kernel.

No. The rank-nullity theorem givesbut this guarantees infinitely many solutions only when the linear system is consistent. For a counterexample takeThere is no solution, since the left side always vanishes.

Yes. A surjective $T_A$ has $\mathrm{im}{T}_A={\mathbb{R}}^n$, so every $\mathbf{b}$ is attained. Its rank of a linear map is $n$, and the rank-nullity theorem givesThe solution set of a linear system is therefore a translate of a nonzero vector subspace, containing infinitely many solutions as in iii.