# Isomorphisms

• Oct 28th 2010, 08:14 AM
mathbeginner
Isomorphisms
(a) Let S : U -> V and T : V -> W be linear maps. Show that Ker(TS) =S^-1(Ker T).

(b) Let S : V -> W be a surjective linear map and M a subspace of W. Show that V/S^-1(M)Isomorphisms W/M.

Hint: Apply part (a) to S : V-> W and Q : W ->W/M.
• Oct 28th 2010, 09:03 PM
Drexel28
Quote:

Originally Posted by mathbeginner
(a) Let S : U -> V and T : V -> W be linear maps. Show that Ker(TS) =S^-1(Ker T).

(b) Let S : V -> W be a surjective linear map and M a subspace of W. Show that V/S^-1(M)Isomorphisms W/M.

Hint: Apply part (a) to S : V-> W and Q : W ->W/M.

Note that if $\displaystyle u\in \ker TS$ then $\displaystyle TSu=0$ or said differenlty $\displaystyle T(Su)=0$ thus $\displaystyle Su\in\ker T$ and so $\displaystyle u\in S^{-1}\left(\ker T\right)$. The other directly is just as easy.

For b), just take the hint. (assuming that $\displaystyle Q$ is the canonical map $\displaystyle Q:V\toV/M:v\mapsto v+M$) We know that $\displaystyle QS:V\to W/M$ is a surjective linear map and by the FIT $\displaystyle V/\left(\ker QS\right)\cong V/M$ but noticing that $\displaystyle \ker QS=S^{-1}(\ker Q)=S^{-1}(M)$ we see by our previous problem that $\displaystyle V/(S^{-1}(M)}=V/\left(\ker QS\right)\cong V/M$