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Homomorphisms and isomorphisms

  1. Mar 16, 2007 #1
    1. The problem statement, all variables and given/known data

    Let[tex]v_1,v_2,...v_n[/tex] be a basis of V and let [tex]w_1,w_2,...w_n[/tex] be any n elements in V. Define T on V by
    [tex](\lambda_1 v_1+\lambda_2 v_1+...+\lambda_n v_n)T=\lambda_1w_1+....\lambda_n w_n.[/tex]
    a)Show that R is a homomorphism of V into itself.
    b)When is T an isomorphism?

    2. Relevant equations
    An isomorphism is a 1-1 mapping.


    3. The attempt at a solution
    I think I have part a) done, although It makes more sense to me to prove that T is a Hom, not R...not sure where the R came from.

    For part b, I am thinking that since the [tex]v_i[/tex] are a basis that the Iso would be true if the w's were a basis also?
    IK know that an iso occurs when the Kernel is zero..but I'm still kinda confused here.
    Any hints or advice will be appreciated.
    Thanks,
    CC
     
  2. jcsd
  3. Mar 16, 2007 #2

    matt grime

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    Homework Helper

    Obviously, in order for T to be an iso, w_i must be a basis - the image lies in the span of the w_i, and that can only be all of V if they are a basis. That says T an iso implies w_i a basis. What about the converse?

    Now, suppose that v is sent to zero by T, i.e. vT=0. Then what does that say about the linear combination of the w_i that v is sent to?
     
    Last edited: Mar 16, 2007
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