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Math Help - Orthogonal complement subspaces

  1. #1
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    Orthogonal complement subspaces

    Prove that:
    ( \bot= orthogonal complement)

    (U \cap W)^\bot  = U^\bot + W^\bot

    Attempt at solution:
    \rightarrow
    Let x \in (U \cap W)^\botand y \in U + W
    Then <x,y> = <x,u> + <x,w> = 0 + 0
    Therefore x \in U^\bot + W^\bot

    \leftarrow
    Let  x \in U^\bot + W^\bot and y \in U \cap W \Rightarrow y \in U and y \in W
    Then <x,y> = <x,u> \cap <x,w> = 0 \cap 0 = 0
    Therfore, x \in (U \cap W)^\bot

    Is this okay or is it totally wrong? Please help. Thanks alot in advance.
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  2. #2
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    Quote Originally Posted by whoadude16 View Post
    Prove that:
    ( \bot= orthogonal complement)

    (U \cap W)^\bot  = U^\bot + W^\bot

    Attempt at solution:
    \rightarrow
    Let x \in (U \cap W)^\botand y \in U + W
    Then <x,y> = <x,u> + <x,w> = 0 + 0

    Make sure here to write y=u+w\,,\,u\in U\,,\,w\in W , and remark you're using additivity of the inner product

    Therefore x \in U^\bot + W^\bot


    As above, write x=u'+w'\,,\,u'\in U^\perp\,,\,w'\in W^\perp , and don't write u,w

    below: it's enough you already remarked that y\in U\cap W

    Tonio


    \leftarrow
    Let  x \in U^\bot + W^\bot and y \in U \cap W \Rightarrow y \in U and y \in W
    Then <x,y> = <x,u> \cap <x,w> = 0 \cap 0 = 0
    Therfore, x \in (U \cap W)^\bot

    Is this okay or is it totally wrong? Please help. Thanks alot in advance.
    .
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  3. #3
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    Quote Originally Posted by whoadude16 View Post
    Prove that:
    ( \bot= orthogonal complement)

    (U \cap W)^\bot  = U^\bot + W^\bot

    Attempt at solution:
    \rightarrow
    Let x \in (U \cap W)^\botand y \in U + W
    Then <x,y> = <x,u> + <x,w> = 0 + 0
    Therefore x \in U^\bot + W^\bot
    Not valid. The fact that two numbers add to 0 doesn't mean the two numbers must each be 0.

    \leftarrow
    Let  x \in U^\bot + W^\bot and y \in U \cap W \Rightarrow y \in U and y \in W
    Then <x,y> = <x,u> \cap <x,w> = 0 \cap 0 = 0
    Therfore, x \in (U \cap W)^\bot

    Is this okay or is it totally wrong? Please help. Thanks alot in advance.
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    Not valid. The fact that two numbers add to 0 doesn't mean the two numbers must each be 0.
    So if i said what tonio suggested, \Rightarrow y = u + w, u \in U, w \in W, will it be okay?

    Prove that:
    (= orthogonal complement)



    Attempt at solution:

    Let and \Rightarrow y = u + w, u \in U, w \in W
    Then using addivity of inner product

    Therefore


    Let \Rightarrow x = u' + w', u' \in U^\perp, w \in W^\perp and y \in U \cap W
    Then <x,y> = <u',y> \cap <w',y> = 0
    Therfore,
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