Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 5 - Inner Product Spaces - 5.2 Inner Product Spaces - 5.2 Exercises - Page 247: 94


$W^\perp$ is spanned by the vectors $\{(-2,1,0),(-3,0,1)\} $.

Work Step by Step

Let $W$ be the subspace spanned by the vector $(1,2,3)$. Then, $W^\perp$ can calculated as follows \begin{aligned} W^{\perp} &=\left\{(x, y, z) \in \mathbb{R}^{3} :(x, y, z) \cdot(1,2,3)=0\right\} \\ &=\left\{(x, y, z) \in \mathbb{R}^{3} : x+2 y+3 z=0\right\} \\ &=\left\{(x, y, z) \in \mathbb{R}^{3} : x=-2 y-3 z\right\} \\ &=\{(-2 y-3 z, y, z) : y, z \in \mathbb{R}\} \\ &=\{(-2 y, y, 0)+(-3 z, 0, z) : y, z \in \mathbb{R}\} \\ &=\{y(-2,1,0)+z(-3,0,1) : y, z \in \mathbb{R}\} \\ &=\operatorname{span}\{(-2,1,0),(-3,0,1)\} \end{aligned} Therefore, $W^\perp$ is spanned by the vectors $\{(-2,1,0),(-3,0,1)\} $.
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