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 - 5.3 Orthonormal Bases: Gram-Schmidt Process - 5.3 Exercises - Page 257: 6


The set $\{(1,2),(-\frac{2}{5},\frac{1}{5})\}$is orthogonal, not orthonormal and it is a basis for $R^2$.

Work Step by Step

given $\{(1,2),(-\frac{2}{5},\frac{1}{5})\}$ let $v_1=(1,2),v_2=(-\frac{2}{5},\frac{1}{5})$ (a) since $v_1v_2=-\frac{2}{5}+\frac{2}{5}=0$ then, the set $\{(1,2),(-\frac{2}{5},\frac{1}{5})\}$ is orthogonal. (b) scince $\left\|{v}_{1}\right\|=\sqrt{v_{1} \cdot v_{1}}=\sqrt{1+4}=\sqrt{13}\neq1$ then, the set $\{(1,2),(-\frac{2}{5},\frac{1}{5})\}$ is not orthonormal. (c) by the corollary to Theorem 5.10, $\{(1,2),(-\frac{2}{5},\frac{1}{5})\}$ is a basis for $R^2$.
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