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Math Help - Proof of intersection and sum of vector spaces

  1. #1
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    Proof of intersection and sum of vector spaces

    Hi,
    how to prove this:
    V^{\bot}\cap W^{\bot}=(V+W)^{\bot}
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  2. #2
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    Quote Originally Posted by lukaszh View Post
    Hi,
    how to prove this:
    V^{\bot}\cap W^{\bot}=(V+W)^{\bot}
    if x \in V^{\bot}\cap W^{\bot}, and v \in V, \ w \in W, then <x,v+w>=<x,v>+<x,w>=0+0=0. thus V^{\bot}\cap W^{\bot} \subseteq (V+W)^{\bot}.

    conversely, we have V \subseteq V+W, and hence (V+W)^{\bot} \subseteq V^{\bot}. similarly (V+W)^{\bot} \subseteq W^{\bot}. thus (V+W)^{\bot} \subseteq V^{\bot}\cap W^{\bot}.
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  3. #3
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    I dont understand this:
    Quote Originally Posted by NonCommAlg View Post
    ... then <x,v+w>=<x,v>+<x,w>=0+0=0. thus V^{\bot}\cap W^{\bot} \subseteq (V+W)^{\bot}....
    How this V^{\bot}\cap W^{\bot} \subseteq (V+W)^{\bot} emerges of <x,v+w>=<x,v>+<x,w>=0+0=0
    Thanks
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  4. #4
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    Quote Originally Posted by lukaszh View Post
    I dont understand this:


    How this V^{\bot}\cap W^{\bot} \subseteq (V+W)^{\bot} emerges of <x,v+w>=<x,v>+<x,w>=0+0=0
    Thanks
    in order to prove that V^{\bot}\cap W^{\bot} \subseteq (V+W)^{\bot}, you need to show that if x \in V^{\bot}\cap W^{\bot}, then x \in (V+W)^{\bot}. to prove that x \in (V+W)^{\bot}, you need to show that x is orthogonal to every

    element of V+W, i.e. <x,z>=0, \ \forall z \in V+W. so you choose an element of z \in V+W. then z=v+w, for some v \in V, w \in W. we have <x,z>=<x,v>+<x,w>=0, because

    x \in V^{\bot}\cap W^{\bot}.
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  5. #5
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    Thank you, it's simple :-)
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