TITLE: Rank of $ T_3$ s.t $ T_3 (T_1)=T_2$
QUESTION [5 upvotes]: Let $T_1,T_2 : R^5 \to R^3$ be linear transformations s.t rank($T_1$)=3 and nullity ($T_2$)=3 . Let $T_3:R^3 \to R^3 $be linear transformation s.t  $ T_3(T_1)=T_2.$ Then find rank of $T_3$

REPLY [7 votes]: By the rank-nullity theorem  we have
$$\operatorname{rank}(T_2)=\dim \Bbb R^5-\operatorname{nullity}(T_2)=2$$
so
$$2=\operatorname{rank}(T_2)=\operatorname{rank}(T_3(T_1))=\dim(T_3(T_1(\Bbb R^5))=\dim(T_3(\Bbb R^3))=\operatorname{rank}(T_3)$$