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Another notation for the dual space

  1. Oct 7, 2007 #1

    quasar987

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    My professor uses the notation Hom(V,[itex]\mathbb{R}[/itex]) for the dual space of V. I usually see V* rather. What does the notation Hom(V,[itex]\mathbb{R}[/itex]) stands for exactly? I suppose the domain of Hom is wider than just (vector spaces) x (their field).

    The notation suggest it is the set of homomorphisms btw V and R, but this would not accurately describe V* because we want the elements of V* to be linear fct, not just group homomorphisms (we need the f(av)=af(v) part of linear too).

    So does the notation comes from a more category theory perspective, where linear maps are considered as "vector space homomorphisms" or something like that?

    Just thoughts. I just wanna know what the general meaning for Hom is. Thx
     
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  3. Oct 7, 2007 #2

    matt grime

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    What is the definition of the dual space that you have that is _not_ homs into the underlying field? Hom, in the category of vector spaces _means_ linear map. It is nothing to do with group homomorphisms.
     
  4. Oct 7, 2007 #3

    quasar987

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    The dual space of V to me is the vector space V* of all linear functions from V to its underlying field.

    It's the first time I see the notation Hom being used for anything at all, so I am asking what it means in the most general context. Thank you.
     
  5. Oct 7, 2007 #4

    Hurkyl

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    Hom(X, Y) is the object of all maps from X to Y.

    The precise meaning of 'object' and 'map' depends on the context. In this case, 'object' means 'vector space' and 'map' means 'linear transformation'.
     
  6. Oct 7, 2007 #5

    Hurkyl

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    I should add...

    Let T : Y->W and S : X->Z be linear.
    Hom(X, T) is the obvious map Hom(X, Y) -> Hom(X, W).
    Hom(S, Y) is the obvious map Hom(Z, Y) -> Hom(X, Y).
    Hom(S, T) is the obvious map Hom(Z, Y) -> Hom(X, W).
     
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