Suppose VVV is finite-dimensional and T1,T2∈L(V,W)T_1,T_2 \in \mathcal L(V,W)T1,T2∈L(V,W). Prove that range T1⊂range T2\text{range }T_1 \subset \text{range }T_2range T1⊂range T2 if and only if there exists S∈L(V,V)S \in \mathcal L(V,V)S∈L(V,V) such that T1=T2ST_1 = T_2ST1=T2S.
TODO