- #1
MushroomPirat
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Homework Statement
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].
Homework Equations
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!