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: 35


(a) $\langle p, q \rangle = 4$ (b) $\| p \| =\sqrt{11}$ (c) $\| q \| = \sqrt{2}$ (d) $d(p,q)= \sqrt{21}$

Work Step by Step

$p(x)=1-x-3x^2, \quad q(x)=x-x^2$ (a) $\langle p, q \rangle = a_0b_0+a_1b_1+a_2b_2=0-1+3=4$ (b) $\| p \| =\sqrt{\langle p, p\rangle}=\sqrt{a_0^2+a_1^2+a_3^2}=\sqrt{1+1+9}=\sqrt{11}$ (c) $\| q \| =\sqrt{\langle q, q\rangle}=\sqrt{b_0^2+b_1^2+b_3^2}=\sqrt{0+1+1}=\sqrt{2}$ (d) $d(p,q)=\| p-q \|=\| 1-2x-4x^2 \|=\sqrt{1+4+16}=\sqrt{21}$
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