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Math Help - Isomorphisms

  1. #1
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    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.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathbeginner View Post
    (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 u\in \ker TS then TSu=0 or said differenlty T(Su)=0 thus Su\in\ker T and so u\in S^{-1}\left(\ker T\right). The other directly is just as easy.

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