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Let $V$ and $W$ be vector spaces over a field $K$ and $T : V \to W$ be a linear map.

Suppose that $T$ is injective. Show that there exists a linear map $S : \text{Im } T \to V$ such that $S\circ T(v) = v$ for all $v\in V$ , and $T\circ S(w) = w$ for all $w \in \text{Im }T$

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Since $T$ is injective then linear map $T\colon V\rightarrow \operatorname{Im} T$ is bijective so let $S=T^{-1}$ and the result follows easily.

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Since $T$ is injective you have that $T:V\rightarrow imT$ is bijective and since $T$ is linear it has an inverse $S$ which has the required property .

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