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

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


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

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

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