Answer
The eigenspace is 1 2 dimensional subspace of $R^2$
Basis is $\Big\{\begin{bmatrix}
3\\
2
\end{bmatrix}
\Big\}$
Work Step by Step
$A-4I=\begin{bmatrix}
10&-9\\
4&-2
\end{bmatrix}-4\begin{bmatrix}
1&0\\
0&1
\end{bmatrix}=\begin{bmatrix}
6&-9\\
4&-6
\end{bmatrix}$
Row reduce augmented matrix for $(A-4I)x=0$
$\begin{bmatrix}
6&-9&0\\
4&-6&0
\end{bmatrix}$~$\begin{bmatrix}
1&-1.5&0\\
0&0&0
\end{bmatrix}$
$x_1=1.5x_2$
$x_2$ is free
The eigenspace is 1 2 dimensional subspace of $R^2$
Basis is $\Big\{\begin{bmatrix}
3\\
2
\end{bmatrix}
\Big\}$
