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: 1


The set $\{(2,-4),(2,1)\}$ orthogonal, orthonormal and it is a basis for $R^2$

Work Step by Step

given $\{(2,-4),(2,1)\}$ let $v_1=(2,-4) , v_2=(2,1)$ (a) since $v_1 v_2=4-4=0$ then, the space $\{(2,-4),(2,1)\}$ is orthogonal. (b) scince $\left\|\mathbf{v}_{1}\right\|=\sqrt{\mathbf{v}_{1} \cdot \mathbf{v}_{1}}=\sqrt{4+16}=\sqrt{20}\neq1$ then, the space {(2,−4),(2,1)} is not orthonormal. (c) by the corollary to Theorem 5.10,$\{(2,-4),(2,1)\}$ is a basis for $R^2$
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