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: 65


see the proof below.

Work Step by Step

Let $f(x)=\cos x, g(x)=\sin x $, $C[-\frac{\pi}{2},\frac{\pi}{2},]$ \begin{aligned}\langle f,g\rangle &=\int_{-\frac{\pi}{2},}^{\frac{\pi}{2}}\cos x \sin x d x\\ &=\int_{-\frac{\pi}{2},}^{\frac{\pi}{2}}\sin x d\sin x \\ &=\left[\frac{1}{2}\sin^2x\right]_{-\frac{\pi}{2},}^{\frac{\pi}{2}} \\ &=0 \end{aligned}. then $f$ and $g$ are orthogonal in the inner product space $C[-\frac{\pi}{2},\frac{\pi}{2},]$.
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