(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose that [itex]W[/itex] is finite dimensional and [itex]T:V\rightarrow W[/itex]. Prove that [itex]T[/itex] is injective if and only if there exists [itex]S:W\rightarrow V[/itex] such that [itex]ST[/itex] is the identity map on [itex]V[/itex].

2. Relevant equations

3. The attempt at a solution

First, suppose that [itex]T[/itex] is injective and let [itex]Tu=Tv[/itex] for [itex]u,v\in V[/itex]. Clearly, [itex]Tu-Tv=0[/itex] and thus [itex]S(Tu-Tv)=0[/itex]. From this, we can see that [itex]STu=STv[/itex]. However, since [itex]T[/itex] is injective, then [itex]u=v[/itex]. Therefore, there exists an [itex]S[/itex] such that [itex]ST[/itex] is the identity map on [itex]V[/itex]. In the other direction, suppose [itex]ST[/itex] is the identity map on [itex]V[/itex], and let [itex]Tu=Tv[/itex]. From the previous argument, we can see that [itex]STu=STv[/itex], and thus [itex]u=v[/itex], so [itex]T[/itex] is injective.

I think the second part of my proof is right, going from identity map to injectivity, but I'm just not sure about my argument for the first half.

Thanks for your help!

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# LA - Identity Maps and Injectivity

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