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Math Help - Vector spaces and homomorphism.

  1. #1
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    Mar 2009
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    Vector spaces and homomorphism.

    1) If V and W are finite dimensional vector spaces over F with
    dimension m and n respectively, then prove that Hom(V,W) is also a
    finite dimensional vector with dimension mn.

    2) If dim V=m, then pt dim Hom(V,V)=m^2

    3) If dim V=m, then pt dim Hom(V,F)=m

    (Hom --> Homomorphism)

    where Hom(V,W)={ T : V --> W, where T is homomorphism }.

    Thanks in advance.
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  2. #2
    Newbie
    Joined
    Apr 2009
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    Hi there!!!

    Let b1,...,bm be a basis for V and let c1,...,cn be a basis for W.
    One way is this:
    define m times n linear maps f_{i,j}: V to W like this:
    f_{i,j} (b_k) = cj if i=k and 0 otherwise.
    Then one can show that the f_{i,j} are linearly independent in HOM(V,W),
    and that each linear map in HOM(V,W) can be written as linear combination
    of the f_{i,j}. That's pretty much it (you need to work out the details a bit).
    The other two statements follow as a corollary
    (recall that F is a one dimensional
    vector space over itself).

    Best,

    ZD
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