Let $V$ and $W$ be vector spaces over a field $F$, $T:V\rightarrow W$ be a linear transformation, and $S:W\rightarrow V$ be a function such that $ST$ is the identity function on $V$ and $TS$ is the identity function on $W$. Prove that $S$ is a linear transformation.

My attempt: Let $\alpha$, $\beta\in V$, $c\in F$. Then $ST(c\alpha+\beta)=c\alpha+\beta=cST\alpha+ST\beta$, where $T\alpha,T\beta\in W$, so $S$ is linear.

Is there a way to do this starting with vectors in $W$? My professor wasn't happy with this solution, but I can't see another way to solve this.

  • 2
    $\begingroup$ You need to argue that $T$ is surjective. Otherwise you have only show that $S$ is linear on the image of $T$. $\endgroup$ – user251257 May 15 '18 at 23:21

Yes, you need to start out from vectors $\alpha, \beta$ in $W$, and use the other assumption $TS=id_W$ as well:

$$S(c\alpha+\beta) =S(cTS(\alpha) +TS(\beta)) =ST(cS(\alpha) +S(\beta)) =cS(\alpha) +S(\beta) $$


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