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Math Help - Direct sum and Isomorphisms

  1. #1
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    Direct sum and Isomorphisms

    Prove that if V = M (Direct sum) N, then
    V/M (Isomorphisms.) N. Hint: Restrict the quotient mapping V /M to N and the kernel and image of the restricted mapping.



    plx help
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  2. #2
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    Quote Originally Posted by mathbeginner View Post
    Prove that if V = M (Direct sum) N, then
    V/M (Isomorphisms.) N. Hint: Restrict the quotient mapping V /M to N and the kernel and image of the restricted mapping.



    plx help


    As any v\in V has a unique expression as v=m_v+n_v\,,\,\,m_v\in M\,,\,n_v\in N , define f:V\to N by f(v=m_v+n_v):= n_v

    Uniqueness of expression gives you that f is well defined, and now just prove that f is an isomorphism of vec. spaces.

    Tonio


    Pd. Of course, f is NOT an isomorphism! You need to find its kernel and use the fist isomorphism theorem
    Last edited by tonio; October 28th 2010 at 09:48 PM. Reason: Correction of mistake
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathbeginner View Post
    Prove that if V = M (Direct sum) N, then
    V/M (Isomorphisms.) N. Hint: Restrict the quotient mapping V /M to N and the kernel and image of the restricted mapping.



    plx help
    Assuming you're working with F.D. vec. spaces if you only need to show that they are isomorphic and you need not exhibit the isomorphism why not note that if \{y_1,\cdots,y_n\} is a basis for \mathcal{N} then y_1+\mathcal{M},\cdots,y_n+\mathcal{M} is a basis for \mathcal{V}/\mathcal{M}. In particular, \dim \mathcal{V}/\mathcal{M}=n=\dim\mathcal{N} and so they are evidently isomorphic.
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  4. #4
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    Would you happen to be a uOttawa student? I have that same question on an assignment of mine
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