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Math Help - orthogonal complement

  1. #1
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    orthogonal complement

    Let W = span \[\left \{ \begin{bmatrix}<br />
1\\ <br />
2\\ <br />
-1<br />
\end{bmatrix}, \begin{bmatrix}<br />
-1\\ <br />
3\\ <br />
2<br />
\end{bmatrix} \right \}\]<br />

    Find a basis for the \[W^{\perp }\]<br />

    I am really confused on how to find this.

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

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

    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
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    Russia
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    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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