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Guenuegoomyns

Answered question

2021-11-21

Find a basis for the set of vectors in $R^{3}$ in the plane x+2y+z=0

Answer & Explanation

Tionant

Beginner2021-11-22Added 17 answers

Solving for $x = - 2 y - z$ , we can plug that into x to get rid on the uneeded variable
$x = - 2 y - z$
$y = y$
$z = z$
Because $x = - 2 y - z$ we can plug that into x to get rid of the uneeded variable.
$[\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} - 2 y - z \\ y \\ z \end{matrix}] = y [\begin{matrix} - 2 \\ 1 \\ 0 \end{matrix}] + z [\begin{matrix} - 1 \\ 0 \\ 1 \end{matrix}]$
We have found spanning set with no unneeded variables, which by the spanning set theorem, is the basis.
$[\begin{matrix} - 2 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} - 1 \\ 0 \\ 1 \end{matrix}]}$

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