# Find a basis for the eigenspace corresponding to each listed

Find a basis for the eigenspace corresponding to each listed eigenvalue of A below.
$A=\begin{bmatrix}1&0\\-1&2\end{bmatrix},\lambda=2,1$

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Donald Proulx

Given matrix A is
$A=\begin{bmatrix}1&0\\-1&2\end{bmatrix}$
f it;s eigen value $$\displaystyle\lambda={2},{1}$$
Next find eigen vector corresponding eigen value.
For $$\displaystyle\lambda={2}$$
$$\displaystyle\Rightarrow{A}{x}=\lambda{x}$$
$\Rightarrow\begin{bmatrix}1&0\\-1&2\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=2\begin{bmatrix}x_1\\x_2\end{bmatrix}$
$$\displaystyle\Rightarrow{x}_{{2}}={2}{x}_{{1}}$$
$$\displaystyle-{x}_{{1}}+{2}{x}_{{2}}={2}{x}_{{2}}$$
$$\displaystyle\Rightarrow{x}_{{1}}-{2}{x}_{{1}}={0}$$
$$\displaystyle\Rightarrow-{x}_{{1}}={0}$$
$$\displaystyle\Rightarrow{x}_{{2}}={1}$$
$\Rightarrow x=\begin{bmatrix}0\\1\end{bmatrix}$
For $$\displaystyle\lambda={1}$$
$$\displaystyle\Rightarrow{A}{x}=\lambda x$$
[\Rightarrow\begin{bmatrix}1&0\\-1&2\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}x_1\\x_2\end{bmatrix}\]
$$\displaystyle\Rightarrow{x}_{{1}}={x}_{{1}}$$
$$\displaystyle-{x}_{{1}}+{2}{x}_{{2}}={x}_{{2}}$$
$$\displaystyle\Rightarrow{x}_{{1}}={x}_{{2}}$$
$x=\begin{bmatrix}1\\1\end{bmatrix}$
is a basis for the eigenspace corresponding to $$\displaystyle\lambda={1}$$