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: 67


see the proof below.

Work Step by Step

Let $f(x)= x, g(x)=\frac{1}{2}(5x^3-3x) $, $C[-1,1]$ \begin{aligned}\langle f,g\rangle &=\int_{-1}^{1}\frac{1}{2}(5x^4-3x^2) d x\\ &=\left[\frac{1}{2}( x^5- x^3) \right]_{-1}^{1} \\ &=0 \end{aligned}. then $f$ and $g$ are orthogonal in the inner product space $C[-1,1]$.
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