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prelimaf1

Answered question

2021-11-21

Find a basis for the eigenspace corresponding to each listed eigenvalue of A below.

A = [\begin{matrix} 1 & 0 \\ - 1 & 2 \end{matrix}], λ = 2, 1

Answer & Explanation

Donald Proulx

Beginner2021-11-22Added 18 answers

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

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