1
$\begingroup$

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.

$\endgroup$
  • 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
2
$\begingroup$

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

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.