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: 77


$$\operatorname{proj}_{{g}} {f} =0.$$

Work Step by Step

Let $f(x)=x, \quad g(x)=1, \quad C[-1,1] , \quad\langle f, g\rangle=\int_{-1}^1 f(x)g(x) d x.$ We have $$\langle f, g\rangle=\int_{-1}^1 x d x=\left[\frac{ 1}{2 } x^2\right]_{-1}^1=0,$$ $$\langle g,g\rangle=\int_{-1}^{1} 1 d x= \left[x\right]_{-1}^{1}=2.$$ Now, the orthogonal projection of $f$ onto $g$ is given by $$\operatorname{proj}_{{g}} {f} =\frac{\langle{f}, {g}\rangle}{\langle{g}, {g}\rangle} {g}=0.$$
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