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Math Help - linear and bijective

  1. #1
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    linear and bijective

    Let V and W be finite-dimensional vector spaces. Show that the mapping
    A: L(V,W) -> L(W*,V*), A(T)=T*

    is linear and bijective.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathbeginner View Post
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    A) Where's the work? This isn't an answer service

    B) Can you define your notation, in particular what T^* is. Is it T^*:W^*\to V^*:\varphi\mapsto \varphi\circ T? If so, I can give you the hint that it's evidently injective and linear (this is for you to answer) and surjetivity follows from a dimension argument (noting that \dim\text{Hom}\left(V,W\right)=\dim\text{Hom}\left  (W^*,V^*\right)
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    A) Where's the work? This isn't an answer service

    B) Can you define your notation, in particular what T^* is. Is it T^*:W^*\to V^*:\varphi\mapsto \varphi\circ T? If so, I can give you the hint that it's evidently injective and linear (this is for you to answer) and surjetivity follows from a dimension argument (noting that \dim\text{Hom}\left(V,W\right)=\dim\text{Hom}\left  (W^*,V^*\right)
    so you are saying if I want to prove that [Math]\varphi\mapsto \varphi\circ T[/tex] is linear. I have to find something that live in \varphi\ and that prove it is linear???
    here is what I am trying to do let g,h in \varphi\ and c,d are scalars
    so (T*(cg+dh))(v) = (A(T)(cg+dh)(v)=....= (cT*g+dT*h)(v)
    Am I get it???

    For proving they are bijective I have to show that dimV=dimW.
    But how can I find the dim V and dim W???
    Last edited by mathbeginner; November 29th 2010 at 02:19 PM.
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  4. #4
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    Quote Originally Posted by Drexel28 View Post
    A) Where's the work? This isn't an answer service

    B) Can you define your notation, in particular what T^* is. Is it T^*:W^*\to V^*:\varphi\mapsto \varphi\circ T? If so, I can give you the hint that it's evidently injective and linear (this is for you to answer) and surjetivity follows from a dimension argument (noting that \dim\text{Hom}\left(V,W\right)=\dim\text{Hom}\left  (W^*,V^*\right)
    if it is surjetivity then the dim V>= dim W but how can I find that dimW?
    as I know that dim V= dimT(V) + Ker (T) but where is T(v) lives?
    Is it live in V or in T this linear map when we apply V then we have W.
    I know that surjective is all V apply linear map T is on W.
    But for me it is so hard to show that everything is on W.
    I have problem on showing that is surjective.
    Plx help.
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