Let $S\colon \mathbb{R}^3\to \mathbb{R}^4$ and $T\colon \mathbb{R}^4\to \mathbb{R}^3$ be two linear transformations. If $TS$ is an identity map then can we say that $ST$ is also an identity map? I think it should be, because $ST\colon \mathbb{R}^3\to \mathbb{R}^3$. Please guide me.
If $S\colon \mathbb{R}^3\to\mathbb{R}^4$ and $T\colon \mathbb{R}^4\to \mathbb{R}^3$ satisfy $TS=\operatorname{id}$, then is $ST=\operatorname{id}$?
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linear-algebra
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1Are you sure you mean $ST: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$? Do you perhaps mean $ST: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}$, or are you using right composition? – 2012-12-26
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0It’s only when the two vector spaces have the same finite dimension that you can say that $TS$ is identity if and only if $ST$ is identity. – 2012-12-26