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Thread: orthogonal complement

  1. #1
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    Jul 2010
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    5

    orthogonal complement

    Let W = span $\displaystyle \[\left \{ \begin{bmatrix}
    1\\
    2\\
    -1
    \end{bmatrix}, \begin{bmatrix}
    -1\\
    3\\
    2
    \end{bmatrix} \right \}\]
    $

    Find a basis for the $\displaystyle \[W^{\perp }\]
    $

    I am really confused on how to find this.

    So i turned it it a matrix and got the rref.
    $\displaystyle \[\begin{bmatrix}
    1 & -1\\
    2 &3 \\
    -1& 2
    \end{bmatrix}\sim \begin{bmatrix}
    1 & 0\\
    0 & 1\\
    0& 0
    \end{bmatrix}\]
    $

    I must've did this wrong because the answer is $\displaystyle \[\begin{bmatrix}
    7/5 \\
    -1/5 \\
    1
    \end{bmatrix}\]
    $

    I understand how they got the 1, but I have no idea how they got 7/5 and -1/5.
    Can someone please explain. Thanks in advance
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
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    Russia
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    1,506
    Thanks
    1
    First of all W^p have only one vector(why?)

    Recall the definition of vectors in W^p...

    Suppose (a,b,c) is in W^p then:

    You will get:

    (1,2-1)(a,b,c)=0
    (-1,3,3)(a,b,c)=0

    A system of two equations with three variables.
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