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Thread: Orthogonal Prjection Question

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    Orthogonal Prjection Question

    Find the matrix for the orthogonal projection onto the subspace of R^3 defined
    by 3x_1 -x_2  + 3x_3
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  2. #2
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    Quote Originally Posted by flaming View Post
    Find the matrix for the orthogonal projection onto the subspace of R^3 defined
    by 3x_1 -x_2  + 3x_3 \color{red}{} = 0.
    The projection P onto the 1-dimensional subspace spanned by the vector \mathbf{v} = (3,-1,3) is given by P\mathbf{x} = \langle\mathbf{x},\mathbf{v}\rangle/\|\mathbf{v}\|^2 = \tfrac1{19}(3x_1-x_2+x_3)\mathbf{v}, where x is the vector (x_1,x_2,x_3). In matrix notation, this looks like P\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix} = \frac1{19}\begin{bmatrix}9x_1-3x_2+9x_3 \\-3x_1+x_2-3x_3 \\9x_1-3x_2+9x_3 \end{bmatrix} = \frac1{19}\begin{bmatrix}9&-3&9 \\-3&1&-3 \\9&-3&9 \end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmat  rix}. So the matrix of P is M_P = \frac1{19}\begin{bmatrix}9&-3&9 \\-3&1&-3 \\9&-3&9 \end{bmatrix}.

    The 2-dimensional subspace given by 3x_1 -x_2  + 3x_3 = 0 is the orthogonal complement of the above 1-dimensional subspace. So the projection onto it will have matrix I-M_P.
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