Originally Posted by

**Dinkydoe** 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.