Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics I / 3 / b / iii / Solution

By i and ii,The domain of a function of $S^{\prime }$ is $\mathrm{im}T$. Its rank-nullity theorem gives $r_{S^{\prime }}+n_{S^{\prime }}=\dim \mathrm{im}T=r_T$. Applying the rank-nullity theorem for $T$ now giveswhich is exactly the rank-nullity theorem for the function composition $ST$.