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Math Help - Hom and vector spaces...

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Hom and vector spaces...

    Prove that if U,W,V are vector spaces above field F, then:

    Hom(U,VxW) ~= Hom(U,V) x Hom(U,W)

    ("~=" - almost equals )


    Thank you!
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  2. #2
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Prove that if U,W,V are vector spaces above field F, then:

    Hom(U,VxW) ~= Hom(U,V) x Hom(U,W)

    ("~=" - almost equals )


    Thank you!
    first of all "~=" doesn't mean "almost equals"! it's \cong, which means "isomorphic as vector spaces". anyway, let \pi_1: V \times W \longrightarrow V, \ \pi_2 : V \times W \longrightarrow W be the projection maps, i.e.

    \pi_1(v,w)=v and \pi_2(v,w)=w, for all v \in V, \ w \in W. now define \varphi: \text{Hom}(U, V \times W) \longrightarrow \text{Hom}(U,V) \times \text{Hom}(U,W) by \varphi(f)=(\pi_1f, \pi_2f) and show that \varphi is an isomorphism.
    Last edited by NonCommAlg; December 15th 2009 at 01:43 PM.
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Can you please write the exact prove, I mean, prove the isomorphism thing?

    I will appreciate this! :-)
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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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    Please someone HELP!
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  5. #5
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Please someone HELP!
    you're not doing anything but yelling for more help! by giving you the map \varphi i've already done most of the work for you. the rest is easy. show that \varphi is well-defined, linear, one-to-one and onto.

    can't you do any of these? if you want me to write all details for you, that is not going to happen. sorry!
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