Find the orthogonal complement of W and give a basis

cleritere39 2021-11-21 Answered
Find the orthogonal complement of W and give a basis for
\[W=\{\begin{bmatrix}x\\ y\\ z\end{bmatrix}:x=\frac{1}{2}t,\ y=-\frac{1}{2},\ z=2t\}\]

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Expert Answer

Walker Funk
Answered 2021-11-22 Author has 1234 answers
So we need to find two linearly independent vectors orthogonal to
\[\begin{bmatrix}\frac{1}{2}\\-\frac{1}{2}\\2\end{bmatrix}\]
\[\begin{bmatrix}1\\-1\\4\end{bmatrix}\] (or to avoid fractions). To do so, we can just use the definition of orthogonality:
\(\displaystyle{\left({x},{y},{z}\right)}\cdot{\left({1},-{1},{4}\right)}={x}-{y}+{4}{z}={0}\)
Let's let x=1 and y=0. Then we see that \(\displaystyle{z}=-{\frac{{{1}}}{{{4}}}}\). So
\[\begin{bmatrix}1\\0\\-\frac{1}{4}\end{bmatrix}\]
is in W. Now let x=0 and y=1. Then \(\displaystyle{z}={\frac{{{1}}}{{{4}}}}\). So
\[\begin{bmatrix}0\\1\\\frac{1}{4}\end{bmatrix}\]
is also in W. Thus
\[W=\{\begin{bmatrix}1\\0\\-\frac{1}{4}\end{bmatrix},\begin{bmatrix}0\\1\\\frac{1}{4}\end{bmatrix}\}\]
where
\[\{\begin{bmatrix}1\\0\\-\frac{1}{4}\end{bmatrix},\begin{bmatrix}0\\1\\\frac{1}{4}\end{bmatrix}\}\]
Is a basis for W
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