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: 4


(a) the set is not orthogonal. (b) the set is not orthonormal. (c) the set is a basis for $R^2$.

Work Step by Step

Let $u=(11,4)$ and $v=(8,-3)$, then we have (a) since $u\cdot v=88-12=76\neq 0$, then the set is not orthogonal. (b) since the set is not orthogonal then it is not orthonormal. (c) to check if it is a basis for $R^2$, consider the combination $$a(11,4)+b(8,-3)=(0,0)$$ then we have the system $$11a+8b=0, \quad 4a-3b=0.$$ The coefficient matrix has non zero determinant and hence the system has the solution $a=0, b=0$. Then the set is linearly independent and $R^2$ has dimension $2$, then it is a basis for $R^2$.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.