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Math Help - Isomorphism of kernel question

  1. #1
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    Isomorphism of kernel question

    Consider the linear transformation T: V (+) W ---> V given by T(v,w) = v. Show that ker(T) is isomorphic to W. [V (+) W is the Cartesian sum or direct sum]

    What does the rank-nullity theorem now give an alternative proof of?
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Isomorphism of kernel question

    if  x=(x_1,x_2) \in Ker(T) means  x_1 = 0_{v} , so Ker(T) =  (0_{v}, w) w \in W
    Since Dim( V \oplus W ) = Dim(V) + Dim(W)
    Rank(T) = Dim(V)
    so Rank(T) + Dim(Ker(T)) = Dim(V) + Dim(W)
    Dim(Ker(T)) = Dim(W)
    All finite dimensional vector spaces of equal dim are isomorphic.

    additionally, you can set up a linear map from Ker(T) (since its a subspace of your vector space),
    G:Ker(T) \to W  (v_1,w_1) \to w_1
    Its easy to see this is both a linear map and a bijection, thus they are isomorphic.
    Last edited by jakncoke; February 20th 2013 at 06:22 PM.
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