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Math Help - Projection matrices in R3

  1. #1
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    Projection matrices in R3

    The question
    a) Find the standard matrix A_3 of the transformation which projects R^3 onto the x - z plane.
    b) Find the standard matrix A_4 of the transformation which projects the x - z plane in R^3 onto R^2.
    c) Find the standard matrix A_5 of the transformation which 'embeds' R^2 as the x - z plane in R^3.

    I'm not sure how to attempt these. Does it have something to do with S=GG^T? I can work out the first one in my head, but I'd like to know the algorithm for solving the questions.

    Any assistance would be truly appreciated.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Projection matrices in R3

    Quote Originally Posted by Glitch View Post
    a) Find the standard matrix A_3 of the transformation which projects R^3 onto the x - z plane.
    There are several general methods, but in this particular case, the projection of (x,y,z) onto the xz plane directly is (x',y',z')=(x,0,z) that is, \begin{bmatrix}{x'}\\{y'}\\{z'} \end{bmatrix}=\begin{bmatrix}{1}&{0}&{0}\\{0}&{0}&  {0}\\{0}&{0}&{1} \end{bmatrix}\begin{bmatrix}{x}\\{y}\\{z} \end{bmatrix} i.e. A_3=\begin{bmatrix}{1}&{0}&{0}\\{0}&{0}&{0}\\{0}&{  0}&{1} \end{bmatrix}
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  3. #3
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    Re: Projection matrices in R3

    Thanks, I worked that much out. It's b) and c) that are giving me the most trouble.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: Projection matrices in R3

    b) (x,y,z)\to (x',y')=(x,z) so \begin{bmatrix}{x'}\\{y'} \end{bmatrix}=\begin{bmatrix}{1}&{0}&{0}\\{0}&{0}&  {1} \end{bmatrix}\begin{bmatrix}{x}\\{y}\\{z} \end{bmatrix} . Then A_4=\begin{bmatrix}{1}&{0}&{0}\\{0}&{0}&{1} \end{bmatrix}
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