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: 13


(a) the set is orthogonal. (b) the set is orthonormal. (c) the set is not a basis for $R^4$.

Work Step by Step

Let $u=(\frac{\sqrt 2}{2},0,0,\frac{\sqrt 2}{2})$ and $v=(0,\frac{\sqrt 2}{2},\frac{\sqrt 2}{2},0), w=(-\frac{1}{2},\frac{1}{2},-\frac{ 1}{2},\frac{1}{2})$, then we have (a) since $$u\cdot v=0, u\cdot w =-\frac{\sqrt{2}}{4}+\frac{\sqrt{2}}{4}= 0, v\cdot w =\frac{\sqrt{2}}{4}-\frac{\sqrt{2}}{4}= 0$$ then the set is orthogonal. (b) since $$\|u\|=\frac{1}{2}+\frac{1}{2}=1 , \|v\|=\frac{2}{4}+\frac{2}{4}=1, $$ $$\|w\|=\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=1, $$ then the set is orthonormal. (c) sine $R^4$ has dimension $4$ and the set has three vectors, then it is not a basis for $R^4$.
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