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Math Help - Subspaces.

  1. #1
    Newbie
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    Subspaces.

    Hey again!
    Sorry, I don't mean to fill up the entire forum, but as I said in my other post, I have an exam and I need all the help I can get. I figured if anyone can answer these for me while I study, bonus! Here's a few questions I could use the answers for.
    Subspaces.-1-2.jpg

    I put them as pictures because they're much more clear that way than if I tried to type them.

    Again, thank you to whoever is willing to help!
    Last edited by mr fantastic; April 7th 2011 at 05:13 PM. Reason: Deleted excess questions.
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  2. #2
    Junior Member
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    For 1:

    x \in U \Rightarrow <x, U^\perp> = 0 \Rightarrow x \in \left( U^\perp\right)^\perp .

    So we have U\subseteq \left( U^\perp\right)^\perp^{(1)}.

    dim_\mathbb R U = dim_\mathbb R U^\perp, dim_\mathbb R U^\perp= dim_\mathbb R  \left( U^\perp\right)^\perp \Rightarrow dim_ \mathbb R U= dim_\mathbb R  \left( U^\perp\right)^\perp ^{(2)}.

    \overset{(1),(2)}{\Longrightarrow} U = \left( U {^\perp}\right)^{\perp}.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    For 2 :

    (i)

    x\in (U+V)^{\perp}\Rightarrow \ldots

    Next step?

    (ii)

    x\in U^{\perp}\cap V^{\perp}\Rightarrow \ldots

    Next step?
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  4. #4
    MHF Contributor

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    two more hints:

    if <x,u> = 0, and <x,v> = 0 then certainly <x,u+v> = 0.

    now can we say that if u is in U, and v is in V, that u,v are in U+V?
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