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 245: 38


(a) $\langle p, q \rangle = -1$ (b) $\| p \| =\sqrt{6}$ (c) $\| q \| =\sqrt{2}$ (d) $d(p,q)=\sqrt{10}$

Work Step by Step

$p(x)=1-2x-x^2, \quad q(x)=x-x^2$ (a) $\langle p, q \rangle = a_0b_0+a_1b_1+a_2b_2=0-2+1=-1$ (b) $\| p \| =\sqrt{\langle p, p\rangle}=\sqrt{a_0^2+a_1^2+a_2^2}=\sqrt{1+4+1}=\sqrt{6}$ (c) $\| q \| =\sqrt{\langle q, q\rangle}=\sqrt{b_0^2+b_1^2+b_2^2}=\sqrt{0+1+1}=\sqrt{2}$ (d) $d(p,q)=\| p-q \|=\| 1-3x \|=\sqrt{1+9+0 }=\sqrt{10}$
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