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

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

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

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

    (c) Is $\displaystyle \sigma$ normal?
    So since $\displaystyle \sigma$ is real I should figure out whether $\displaystyle \sigma\sigma^T = \sigma^T\sigma$. I guess I should find $\displaystyle \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 $\displaystyle \sigma:\mathbb{R}^3\to \mathbb{R}^3$ be the projection on the plane $\displaystyle V: x-2y+z=0 $

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

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

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

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

    Any help is appreciated.

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

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

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

    Tonio
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