(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.

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- October 28th 2010, 08:14 AMmathbeginnerIsomorphisms
(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. - October 28th 2010, 09:03 PMDrexel28
Note that if then or said differenlty thus and so . The other directly is just as easy.

For b), just take the hint. (assuming that is the canonical map ) We know that is a surjective linear map and by the FIT but noticing that we see by our previous problem that