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Math Help - Help with projection matrix!

  1. #1
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    Help with projection matrix!

    Let V subspace of R3 with inner product V={(x,y,z) in R3 : x+z=0, y=2x}

    1. how can we find the projection P onto V

    2. how can we find the projection matrix P as of the normal basis of R3

    3. how can we find a basis of R3 for which the matrix P will be \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)


    Thanks in advance!!
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  2. #2
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    Quote Originally Posted by ypatia View Post
    Let V subspace of R3 with inner product V={(x,y,z) in R3 : x+z=0, y=2x}

    1. how can we find the projection P onto V

    2. how can we find the projection matrix P as of the normal basis of R3

    3. how can we find a basis of R3 for which the matrix P will be \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)


    Thanks in advance!!
    Clearly, we have a \beta=\{(1,2,-1)\} a basis for \mathsf{V}, and \mathsf{T}:\mathbb{R}^3\to\mathbb{R}^3 with \mathsf{T}((x,y,z))=(x,2x,-x) a projection. \mathsf{T} is probably not an orthogonal projection, however.

    If we want to normalize this, the set \gamma=\{(\sqrt{6}/6,\sqrt{6}/3,-\sqrt{6}/6)\} is a normal basis for \mathsf{V}. For ease of notation, let \alpha refer to the standard basis of \mathbb{R}^3. Then

    [\mathsf{U}]^{\gamma}_{\alpha}=\left(\begin{array}{ccc}\sqrt{6  }/6&0&0\\\sqrt{6}/3&0&0\\-\sqrt{6}/6&0&0\end{array}\right)

    where \mathsf{U}:\mathbb{R}^3\to\mathsf{V} has the same definition as \mathsf{T}, other than its range. So [\mathsf{U}]^{\gamma}_{\alpha} is the projection matrix of \mathsf{T}. Please note again that this is probably not orthogonal.

    However, the matrix

    P=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)

    will never be a projection matrix for any projection of \mathbb{R}^3 along \mathsf{V}. For consider (x,y,z)\in\mathbb{R}^3 with x\neq 0. Then P\cdot (x,y,z)^T=(x,0,0)^T. Since x\neq 0, we have 2x,-x\neq 0, so (x,0,0)\neq (x,2x,-x) \Rightarrow (x,0,0)\notin \mathsf{V}.
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