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
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2Possible duplicate of Rank of $ T_1T_2$ – TASPlasma Dec 09 '16 at 07:23
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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?
TASPlasma
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T$_{1}$is surjective $\Longrightarrow$$\mathbb{\textrm{R}}$$\left(T\right)$= $\mathbb{R}$$^{n}$
T$_{2}$is injective $\Longrightarrow$N$\left(T\right)$=0 $\Longrightarrow0$
- R$\left(T\right)$=Dim$\left(\mathbb{R}^{m}\right)$
rank of T$_{1}$=n
rank of T$_{2}$=m
– Kislay Tripathi Dec 10 '16 at 04:14