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: 96


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Work Step by Step

(a) We have to check the following properties; For any vectors $u,v,w$ in the given space and $ k \in {R}$; (1) $\langle u,u\rangle\geq0$ and $\langle u,u\rangle= 0$ if and only if $u=0$. (2) $\langle u,v\rangle=\langle v,u\rangle$. (3) $\langle k u,v\rangle=k\langle u,v\rangle$. (4) $\langle u+v,w\rangle=\langle u,w\rangle+\langle v,w\rangle$. (b) The orthogonal projection of $u$ onto $v$ is given by $$\operatorname{proj}_{{v}} {u} =\frac{\langle{u}, {v}\rangle}{\langle{v}, {v}\rangle} {v}$$
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