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Math Help - Orthogonal projection

  1. #1
    Newbie
    Joined
    Jul 2010
    Posts
    5

    Orthogonal projection

    Find the projwv for the given vector v and the subspace W.

    Let V be the Euclidean space R^4 and W the subspace with the basis
    \[\begin{bmatrix}<br />
1 & 1 &0  &1 <br />
\end{bmatrix},\begin{bmatrix}<br />
0 & 1 &1 &0 <br />
\end{bmatrix},\begin{bmatrix}<br />
-1 & 0 &0  &1 <br />
\end{bmatrix}\]<br />

    \[v=\begin{bmatrix}<br />
2 & 1 &3 &0<br />
\end{bmatrix}\]<br />

    My work:
    \[proj_{w}v= \frac{<v,w_{1}>}{<w_{1},w_{1}>}w_{1}+\frac{<v,w_{2  }>}{<w_{2},w_{2}>}w_{2}+\frac{<v,w_{3}>}{<w_{3},w_  {3}>}w_{3}\]<br />

    \[= \frac{\begin{bmatrix}<br />
2\\ <br />
1\\ <br />
3\\<br />
0 <br />
\end{bmatrix}\cdot\begin{bmatrix}<br />
1\\ <br />
1\\ <br />
0\\ <br />
1<br />
\end{bmatrix}}{\begin{bmatrix}<br />
1\\ <br />
1\\ <br />
0\\<br />
1 <br />
\end{bmatrix}\cdot\begin{bmatrix}<br />
1\\ <br />
1\\ <br />
0\\ <br />
1<br />
\end{bmatrix}}\begin{bmatrix}<br />
1\\ <br />
1\\ <br />
0\\<br />
1 <br />
\end{bmatrix}+\frac{\begin{bmatrix}<br />
2\\ <br />
1\\ <br />
3\\<br />
0 <br />
\end{bmatrix}\cdot\begin{bmatrix}<br />
0\\ <br />
1\\ <br />
1\\ <br />
0<br />
\end{bmatrix}}{\begin{bmatrix}<br />
0\\ <br />
1\\ <br />
1\\ <br />
0<br />
\end{bmatrix}\cdot\begin{bmatrix}<br />
0\\ <br />
1\\ <br />
1\\ <br />
0<br />
\end{bmatrix}}\begin{bmatrix}<br />
0\\ <br />
1\\ <br />
1\\ <br />
0 <br />
\end{bmatrix}+\frac{\begin{bmatrix}<br />
2\\ <br />
1\\ <br />
3\\<br />
0 <br />
\end{bmatrix}\cdot\begin{bmatrix}<br />
-1\\ <br />
0\\ <br />
0\\ <br />
1 <br />
\end{bmatrix}}{\begin{bmatrix}<br />
-1\\ <br />
0\\ <br />
0\\ <br />
1 <br />
\end{bmatrix}\cdot\begin{bmatrix}<br />
-1\\ <br />
0\\ <br />
0\\ <br />
1 <br />
\end{bmatrix}}\begin{bmatrix}<br />
-1\\ <br />
0\\ <br />
0\\ <br />
1 <br />
\end{bmatrix}\]<br />

    \[= \frac{3}{3}\begin{bmatrix}<br />
1\\ <br />
1\\ <br />
0\\<br />
1 <br />
\end{bmatrix}+\frac{4}{2}\begin{bmatrix}<br />
0\\ <br />
1\\ <br />
1\\ <br />
0 <br />
\end{bmatrix}+\frac{-2}{2}\begin{bmatrix}<br />
-1\\ <br />
0\\ <br />
0\\ <br />
1 <br />
\end{bmatrix}= \begin{bmatrix}<br />
1\\ <br />
1\\ <br />
0\\<br />
1 <br />
\end{bmatrix}+\begin{bmatrix}<br />
0\\ <br />
2\\ <br />
2\\ <br />
0 <br />
\end{bmatrix}+\begin{bmatrix}<br />
1\\ <br />
0\\ <br />
0\\ <br />
-1 <br />
\end{bmatrix}=\]<br />

    I stopped there because my result was no where near the correct answer :-/
    The correct answer is \[\begin{bmatrix}<br />
7/5 &11/5  & 9/5 &-3/5 <br />
\end{bmatrix}\]<br />

    Can someone please tell me what I did wrong. Thanks in advance.
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  2. #2
    MHF Contributor
    Joined
    Aug 2007
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    USA
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    Thanks
    2
    Is your Basis Orthonormal?
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