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Math Help - projection on a plane

  1. #1
    Senior Member Dinkydoe's Avatar
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    projection on a plane

    I'm preparing for a test and somewhat noticed I don't understand everything as well as I thought I did.

    I'm trying to solve the following problem:

    Let \sigma:\mathbb{R}^3\to \mathbb{R}^3 be the projection on the plane V: x-2y+z=0

    (a) Give a basis of \mathbb{R}^3 consisting of eigenvectors of \sigma

    Since (\sigma-I)v = 0 for all v in V, I guess we can take two orthogonal vectors v_1,v_2\in V. But how do we get a third eigenvector? Just taking a third orthogonal vector?

    (b) Give the matrix of \sigma with respect to the standard-basis in \mathbb{R}^3

    (c) Is \sigma normal?
    So since \sigma is real I should figure out whether \sigma\sigma^T = \sigma^T\sigma. I guess I should find  \sigma first.

    Any help is appreciated.
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  2. #2
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    Quote Originally Posted by Dinkydoe View Post
    I'm preparing for a test and somewhat noticed I don't understand everything as well as I thought I did.

    I'm trying to solve the following problem:

    Let \sigma:\mathbb{R}^3\to \mathbb{R}^3 be the projection on the plane V: x-2y+z=0

    (a) Give a basis of \mathbb{R}^3 consisting of eigenvectors of \sigma

    Since (\sigma-I)v = 0 for all v in V, I guess we can take two orthogonal vectors v_1,v_2\in V. But how do we get a third eigenvector? Just taking a third orthogonal vector?

    (b) Give the matrix of \sigma with respect to the standard-basis in \mathbb{R}^3

    (c) Is \sigma normal?
    So since \sigma is real I should figure out whether \sigma\sigma^T = \sigma^T\sigma. I guess I should find  \sigma first.

    Any help is appreciated.

    (1) Find an orthonormal basis for the plane x-2y+z=0 , say \{u_1,u_2\} (use here Gram-Schmidt with any basis of the plane)

    (2) For any v=\begin{pmatrix}x\\y\\z \end{pmatrix} \in\mathbb{R}^3, its (orthogonal) projection on the plane is \sigma(v):=<v,u_1>u_1+<v,u_2>u_2 , with <,> the standard euclidean inner product in \mathbb{R}^3

    (3) From the above get the matrix for \sigma wrt the standard basis of \mathbb{R}^3 and check that indeed \sigma^2=\sigma

    Tonio
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