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Math Help - How to show this isomorphism?

  1. #1
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    How to show this isomorphism?

    Given A \in L(V, W). How can I show that the  V/kerA qutioent space is isomorphic to im A? Thank you very much!
    Last edited by gotmejerry; December 4th 2011 at 09:11 AM.
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  2. #2
    MHF Contributor Amer's Avatar
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    Re: How to show this isomorphism?

    what is the Group L(V,W) ??
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  3. #3
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    Re: How to show this isomorphism?

    L(V, W) represents all linear transformations from V to W
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  4. #4
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    Re: How to show this isomorphism?

    Then A is a linear transformation while V/ker(A) is a vector space. What do you mean by an isomorphism between a linear transformation and a vector space?

    I suspect you mean an isomorphism between the image of V under A and V/ker(A).
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  5. #5
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    Re: How to show this isomorphism?

    Yes, sorry I meant that
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Re: How to show this isomorphism?

    Quote Originally Posted by gotmejerry View Post
    Given A \in L(V, W). How can I show that the  V/kerA qutioent space is isomorphic to im A? Thank you very much!
    This is just the first isomorphism theorem for vector spaces. Try showing that V/\ker A\to\text{im }A: x+\ker A\mapsto A(x) is a well-defined isomorphism. Now (and I mention this only because [despite the strangeness] some classes do this FIRST) if you know the rank-nullity theorem and you're in finite dimensions you know that \dim V=\dim \ker A+\dim\text{im }A and so \dim \text{im }A=\dim V-\dim \ker A=\dim V/\ker A and so the isomorphism follows.
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  7. #7
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    Re: How to show this isomorphism?

    if you start with a basis \{u_1,\dots,u_k\}, of ker(A) you can extend this to a basis \{u_1,\dots,u_k,v_{k+1},\dots,v_n\} for V.

    prove that \{v_{k+1} + \text{ker}(A),\dots,v_n + \text{ker}(A)\} is a basis for V/ker(A), and that \{A(v_{k+1}),\dots,A(v_n)\}

    is a basis for im(A) (this proves the rank-nullity theorem, and gives you drexel28's isomorphism, at the same time).
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