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: 21


$$[x]_B=\left[ \frac{4\sqrt{13}}{13} \quad \frac{7\sqrt{13}}{13}\right]^T.$$

Work Step by Step

Let $x=(1,2)$ and $B=\{ (-\frac{2\sqrt{13}}{13},\frac{3\sqrt{13}}{13}),(\frac{3\sqrt{13}}{13},\frac{2\sqrt{13}}{13}) \}$ To find the coordinates of $x$ relative to $B$, we have to find the following; $$\langle (1,2),(-\frac{2\sqrt{13}}{13},\frac{3\sqrt{13}}{13})\rangle =-\frac{2\sqrt{13}}{13}+\frac{6\sqrt{13}}{13}=\frac{4\sqrt{13}}{13}$$ $$\langle (1,2),(\frac{3\sqrt{13}}{13},\frac{2\sqrt{13}}{13})\rangle =\frac{3\sqrt{13}}{13}+\frac{4\sqrt{13}}{13} =\frac{7\sqrt{13}}{13}.$$ Then, the coordinate matrix of $x$ relative to $B$ is given by $$[x]_B=\left[ \frac{4\sqrt{13}}{13} \quad \frac{7\sqrt{13}}{13}\right]^T.$$
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