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

Since $S^{\prime }$ is the restriction of $S$ to $\mathrm{im}T$,This image of a linear map is contained in $\mathrm{im}S$, so $r_{ST}\le r_S$. Also, applying $S$ to a basis of $\mathrm{im}T$ gives a spanning set for $\mathrm{im}{S}^{\prime }$, of size $r_T$. Thus $r_{ST}\le r_T$, and