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Math Help - Orthogonal projection onto span of vectors using weighted inner product

  1. #1
    Newbie JCS007's Avatar
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    Post Orthogonal projection onto span of vectors using weighted inner product

    Find the orthogonal projection of
     v=\begin{pmatrix}1 \\ 2 \\ -1  \\ 2 \end{pmatrix}
    onto the span of \begin{pmatrix}1 \\ -1 \\ 2  \\ 5 \end{pmatrix} and \begin{pmatrix}2 \\ 1 \\ 0 \\ -1 \end {pmatrix}
    using the weighted inner product <v,w>=4v_1w_1+3v_2w_2+2v_3w_3+v_4w_4
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by JCS007 View Post
    Find the orthogonal projection of
     v=\begin{pmatrix}1 \\ 2 \\ -1 \\ 2 \end{pmatrix}
    onto the span of \begin{pmatrix}1 \\ -1 \\ 2 \\ 5 \end{pmatrix} and \begin{pmatrix}2 \\ 1 \\ 0 \\ -1 \end {pmatrix}
    using the weighted inner product <v,w>=4v_1w_1+3v_2w_2+2v_3w_3+v_4w_4
    Let:

    b_1=\begin{pmatrix}1 \\ -1 \\ 2 \\ 5 \end{pmatrix}

    and:

    b_2=\begin{pmatrix}2 \\ 1 \\ 0 \\ -1 \end {pmatrix}

    and:

     <br />
u_1=\frac{b_1}{||b_1||}<br />

     <br />
u_2=\frac{b_2-\langle b_2,u_1 \rangle u_1}{||b_2-\langle b_2,u_1 \rangle u_1||}<br />

    Then the orthogonal projection is:

    p=\langle v,u_1\rangle u_1 + \langle v,u_2 \rangle u_2

    Where ||x||=\langle x,x \rangle^{\frac{1}{2}}

    RonL
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