Rank of matrix of linear transformations if we know that $T_1T_2$ is bijective

Question

For, $n\neq m$,Let $T_1\colon\mathbb{R}^n \rightarrow \mathbb R^m$ and $T_2 \colon \mathbb R^m \rightarrow \mathbb R^n$ be linear transformation such that $T_{1}T_{2}$ is bijective then. What is the rank of $T_1$ and $T_2$? MY approach: one book says R(T$_{1}$)=R(T$_{2})=m$.another book says R(T$_{1}$)=n and R(T$_{2}$)=m I beleive former is correct .I just need a verification

Answer 1

Hint: It can be proved that if $T_1 \circ T_2$ is bijective, then $T_1$ is surjective, and $T_2$ is injective. Then using the Rank-Nullity Theorem, what can you conclude?