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Algebriac Structures,hom wrecker

  1. Feb 5, 2007 #1
    1. The problem statement, all variables and given/known data

    Let [tex]T:V \rightarrow W[/tex] be a homomorphism. Using T, define a homomorphism [tex]T^*:Hom(W,F) \rightarrow Hom(V,F) [/tex].




    2. Relevant equations



    3. The attempt at a solution

    This is what I have so far:
    let [tex] f \in Hom(W,F)[/tex]
    Define [tex] f(T^*):V \rightarrow F[/tex]
    and [tex](v)((f)T^*)=((v)T)f[/tex]

    Ok so then I need to show that [tex]T^*[/tex] is acutually a homomorphism.
    So I tried this:
    Let [tex]\lambda \in T^*[/tex]
    Then
    [tex]\lambda(f+g)T^*=\lambda(f(T^*)+g(T^*))[/tex]
    [tex]=\lambda((f(T^*)) + \lambda(g(T^*))[/tex]
    [tex]=(\lambda f)(T^*) + (\lambda g)(T^*)[/tex]
    [tex]=(\lambda f +\lambda g) T^*[/tex]
    [tex]=(\lambda f)T^* = (\lambda g)T^*[/tex]

    So [tex]T^*[/tex] is a homomorphism.

    I'm not sure if this is the correct approach since I have a slippery grasp on this stuff. My Prof says I need to also show that [tex]f(T^*) \in Hom(V,F)[/tex] that doesn't seem intuituvely difficult, but my problem is WRITING IT DOWN.

    Any input will be greatly appreciated.

    CC
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Feb 5, 2007
  2. jcsd
  3. Feb 5, 2007 #2

    HallsofIvy

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    Precisely WHAT algebraic structures are V and W and what is F? My guess is that V and W are vector spaces over field F but I don't know for sure.
     
  4. Feb 5, 2007 #3
    Your are correct, V and W are vector spaces over a field F.
    Sorry.
     
  5. Feb 5, 2007 #4

    matt grime

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    Poor lad, having to write things on the wrong side like that.

    I would say to your professor that it is bloody trivial bby definition that fT* is in Hom(V,F): it is a linear map from V to F.
     
  6. Feb 5, 2007 #5
    yeah, I dunno why we are putting things on the right like that, but we do.
    I agree that [tex]f(T^*) \in Hom(V,F) [/tex] is trivial, but apparantly he wants it.
    Am I correct in my attempt at the other part?

    Thanks

    CC
     
  7. Feb 5, 2007 #6

    matt grime

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    You're putting things on the right because you're looking at right modules for rings. There is a good reason for this, but not an interesting one for you.

    fT* is in Hom(V,F) *by definition*. I.e. you defined it as a function from V to F, and the rest of you proof shows it is a linear map i.e. an element of Hom(V,F). Thus *my* response to you professor is "we define a function from V to F as follows: if v is in V then vfT*=(vT)f (IF YOU WANT MORE DETAILS, THEN vT IS IN V SINCE V IS A FUNCTION FROM V TO V, AND f IS A FUNCTION FROM V TO F. Now we show that this is actually a linear map. INSERT YOUR PROOF"

    the caps aren't 'shouty'. You should think about replacing them with something more appropriate. Oh, and don't copy the rest of this into your work either. A professor can tell precisely what you write from a mathematician's writing. Put it in your language.
     
  8. Feb 5, 2007 #7
    Hey,
    Thanks for the response.
    Just so you guys know, My Prof knows me well enough to spot ANYTHING that's not precisely in my own language, so no worries.

    Am I correct in my attempt that [tex]T^*[/tex] is a Hom above? I think since noone has said I'm completely wrong that I must be on the right track.

    Thanks again,
    CC
     
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