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: 22


$$[x]_B=\left[ \sqrt{5} \quad 2\sqrt{5}\right]^T.$$

Work Step by Step

Let $x=(-3,4)$ and $B=\{ (\frac{\sqrt{5}}{5},\frac{2\sqrt{5}}{5}),(-\frac{2\sqrt{5}}{5},\frac{\sqrt{5}}{5}) \}$ To find the coordinates of $x$ relative to $B$, we have to find the following; $$\langle (-3,4), (\frac{\sqrt{5}}{5},\frac{2\sqrt{5}}{5})\rangle =-\frac{3\sqrt{5}}{5}+\frac{8\sqrt{5}}{5}= \sqrt{5} $$ $$\langle (-3,4),(-\frac{2\sqrt{5}}{5},\frac{\sqrt{5}}{5})\rangle =\frac{6\sqrt{5}}{5}+\frac{4\sqrt{5}}{5} =2\sqrt{5}.$$ Then, the coordinate matrix of $x$ relative to $B$ is given by $$[x]_B=\left[ \sqrt{5} \quad 2\sqrt{5}\right]^T.$$
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