prelimaf1

2021-11-21

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

Donald Proulx

Given matrix A is
$A=\left[\begin{array}{cc}1& 0\\ -1& 2\end{array}\right]$
f it;s eigen value $\lambda =2,1$
Next find eigen vector corresponding eigen value.
For $\lambda =2$
$⇒Ax=\lambda x$
$⇒\left[\begin{array}{cc}1& 0\\ -1& 2\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]=2\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]$
$⇒{x}_{2}=2{x}_{1}$
$-{x}_{1}+2{x}_{2}=2{x}_{2}$
$⇒{x}_{1}-2{x}_{1}=0$
$⇒-{x}_{1}=0$
$⇒{x}_{2}=1$
$⇒x=\left[\begin{array}{c}0\\ 1\end{array}\right]$
For $\lambda =1$
$⇒Ax=\lambda x$
$\left[\begin{array}{cc}1& 0\\ -1& 2\end{array}\right]$$\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]$=$\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]$
$⇒{x}_{1}={x}_{1}$
$-{x}_{1}+2{x}_{2}={x}_{2}$
$⇒{x}_{1}={x}_{2}$
$x=\left[\begin{array}{c}1\\ 1\end{array}\right]$
is a basis for the eigenspace corresponding to $\lambda =1$

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