Find the eigenvalues and a basis for each eigenspace of matrix A:
\begin{bmatrix} 1 & -3 & 3 \\ 2 & -2 & 2 \\ 2 & 0 & 0 \\ \end{bmatrix}
I found the eigenvalues by computing $|A-\lambda I|$:
$\lambda_1 = 0,$ $\lambda_2 = 1,$ $\lambda_3 = -2$
How do I find a basis for each eigenspace of matrix A?
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I tried the following:
$\lambda = 0:$
\begin{bmatrix} 1 & -3 & 3 \\ 2 & -2 & 2 \\ 2 & 0 & 0 \\ \end{bmatrix}
Do reduced row echelon form:
\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ \end{bmatrix}
Does this mean one of the results is \begin{bmatrix} 0\\ t\\ t\\ \end{bmatrix}
and to find the other two answers, do the same thing except set $\lambda$ equal to the other two values? Or does finding the basis for each eigenspace of matrix A mean something different?