= Nullity of a composition
{title2=$n(ST)=n(T)+\dim(\operatorname{im}T\cap\ker S)$}
For finite-dimensional <linear maps> $T:U\to V$ and $S:V\to W$, the restriction of $T$ to $\ker(ST)$ maps onto $\operatorname{im}T\cap\ker S$ and has kernel $\ker T$. The <rank-nullity theorem> on this restriction proves the title identity. Since the intersection has dimension at most $\dim\ker S$, it implies $n(ST)\leq n(T)+n(S)$. Equivalently, $r(ST)\geq r(T)+r(S)-\dim V$, the <rank inequality for a composition>. Also $\operatorname{im}(ST)=S(\operatorname{im}T)$ directly gives the upper bounds $r(ST)\leq\min(r(S),r(T))$.
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