<s>[INST] Solving a system of linear equations to find two eigenvectors. So I have
\begin{align*}
x - 2y + z & = 0 \\
-2x + 4y - 2z & = 0 \\
x - 2y + z & = 0
\end{align*}
I know I need to find two eigenvectors for the eigenspace with eigenvalue 2 as I know the matrix is diagonalizable and I've already found the eigenvector for the eigenspace with eigenvalue 8...
I can't seem to solve this. Please help!
 [/INST] $$ \begin{align*}
x - 2y + z & = 0 \\
-2x + 4y - 2z & = 0 \\
x - 2y + z & = 0 
\end{align*} $$
$$ \implies  \begin{align*}
x - 2y + z & = 0 \\
0 & = 0 \\
0 & = 0.
\end{align*}$$
So you have two free variables. Assuming $y=t, z=s,\, t,s\in \mathbb{R} $, then the solution is given by
$$ \begin{bmatrix}
       x  \\
       y\\
       z\end{bmatrix} =  \begin{bmatrix}
       2t-s  \\
       t\\
       s\end{bmatrix} = \begin{bmatrix}
       2t  \\
       t\\
       0\end{bmatrix} +  \begin{bmatrix}
       -s  \\
       0\\
       s\end{bmatrix} = t\begin{bmatrix}
       2  \\
       1\\
       0\end{bmatrix} +  s\begin{bmatrix}
       -1  \\
       0\\
       1\end{bmatrix}.$$
Can you see the two eigenvectors now?
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