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Thread: 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
    $\displaystyle \[\begin{bmatrix}
    1 & 1 &0 &1
    \end{bmatrix},\begin{bmatrix}
    0 & 1 &1 &0
    \end{bmatrix},\begin{bmatrix}
    -1 & 0 &0 &1
    \end{bmatrix}\]
    $

    $\displaystyle \[v=\begin{bmatrix}
    2 & 1 &3 &0
    \end{bmatrix}\]
    $

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

    $\displaystyle \[= \frac{\begin{bmatrix}
    2\\
    1\\
    3\\
    0
    \end{bmatrix}\cdot\begin{bmatrix}
    1\\
    1\\
    0\\
    1
    \end{bmatrix}}{\begin{bmatrix}
    1\\
    1\\
    0\\
    1
    \end{bmatrix}\cdot\begin{bmatrix}
    1\\
    1\\
    0\\
    1
    \end{bmatrix}}\begin{bmatrix}
    1\\
    1\\
    0\\
    1
    \end{bmatrix}+\frac{\begin{bmatrix}
    2\\
    1\\
    3\\
    0
    \end{bmatrix}\cdot\begin{bmatrix}
    0\\
    1\\
    1\\
    0
    \end{bmatrix}}{\begin{bmatrix}
    0\\
    1\\
    1\\
    0
    \end{bmatrix}\cdot\begin{bmatrix}
    0\\
    1\\
    1\\
    0
    \end{bmatrix}}\begin{bmatrix}
    0\\
    1\\
    1\\
    0
    \end{bmatrix}+\frac{\begin{bmatrix}
    2\\
    1\\
    3\\
    0
    \end{bmatrix}\cdot\begin{bmatrix}
    -1\\
    0\\
    0\\
    1
    \end{bmatrix}}{\begin{bmatrix}
    -1\\
    0\\
    0\\
    1
    \end{bmatrix}\cdot\begin{bmatrix}
    -1\\
    0\\
    0\\
    1
    \end{bmatrix}}\begin{bmatrix}
    -1\\
    0\\
    0\\
    1
    \end{bmatrix}\]
    $

    $\displaystyle \[= \frac{3}{3}\begin{bmatrix}
    1\\
    1\\
    0\\
    1
    \end{bmatrix}+\frac{4}{2}\begin{bmatrix}
    0\\
    1\\
    1\\
    0
    \end{bmatrix}+\frac{-2}{2}\begin{bmatrix}
    -1\\
    0\\
    0\\
    1
    \end{bmatrix}= \begin{bmatrix}
    1\\
    1\\
    0\\
    1
    \end{bmatrix}+\begin{bmatrix}
    0\\
    2\\
    2\\
    0
    \end{bmatrix}+\begin{bmatrix}
    1\\
    0\\
    0\\
    -1
    \end{bmatrix}=\]
    $

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

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