GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 1

Question

Let $T_{1}, T_{2}: R^{5} \rightarrow R^{3}$ be linear transformations s.t $\operatorname{rank}\left(T_{1}\right)=3$ and nullity $\left(T_{2}\right)=3$. Let $T_{3}: R^{3} \rightarrow R^{3}$ be linear transformation s.t $T_{3}\left(T_{1}\right)=T_{2}$. Then find rank of $T_{3}$

Answer 1

By the rank-nullity theorem we have
$$
\operatorname{rank}\left(T_{2}\right)=\operatorname{dim} \mathbb{R}^{5}-\operatorname{nullity}\left(T_{2}\right)=2
$$
so
$$
2=\operatorname{rank}\left(T_{2}\right)=\operatorname{rank}\left(T_{3}\left(T_{1}\right)\right)=\operatorname{dim}\left(T_{3}\left(T_{1}\left(\mathbb{R}^{5}\right)\right)=\operatorname{dim}\left(T_{3}\left(\mathbb{R}^{3}\right)\right)=\operatorname{rank}\left(T_{3}\right)\right.
$$