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 246: 76

Answer

(a) The orthogonal projection of $u$ onto $v$ is given by $$\operatorname{proj}_{{v}} {u} = \frac{-19}{22}(4,-1,2,-1)$$ (b) The orthogonal projection of $v$ onto $u$ is given by $$\operatorname{proj}_{{u}} {v} = \frac{-19}{40}(-1,4,-2,3).$$

Work Step by Step

Let $u=(-1,4,-2,3)$, $v=(4,-1,2,-1)$, $\langle{u}, {v}\rangle=u\cdot v$. Then, we have $$\langle{u}, {v}\rangle=-19, \quad \langle{u}, {u}\rangle=40, \quad \langle{v}, {v}\rangle=22.$$ (a) The orthogonal projection of $u$ onto $v$ is given by $$\operatorname{proj}_{{v}} {u} =\frac{\langle{u}, {v}\rangle}{\langle{v}, {v}\rangle} {v}=\frac{-19}{22}(4,-1,2,-1)$$ (b) The orthogonal projection of $v$ onto $u$ is given by $$\operatorname{proj}_{{u}} {v} =\frac{\langle{u}, {v}\rangle}{\langle{u}, {u}\rangle} {u}=\frac{-19}{40}(-1,4,-2,3).$$
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