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: 25


$$[x]_B=\left[ 11 \quad 2 \quad 15\right]^T.$$

Work Step by Step

Let $x=(5,10,15)$ and $B=\{ (\frac{3}{5},\frac{4}{5},0),(-\frac{4}{5},\frac{3}{5},0),(0,0,1) \}$ To find the coordinates of $x$ relative to $B$, we have to find the following; $$\langle (5,10,15),(\frac{3}{5},\frac{4}{5},0)\rangle = 3+8=11$$ $$\langle (5,10,15),(-\frac{4}{5},\frac{3}{5},0)\rangle = -4+6=2$$ $$\langle (5,10,15),(0,0,1)\rangle = 15.$$ Then, the coordinate matrix of $x$ relative to $B$ is given by $$[x]_B=\left[ 11 \quad 2 \quad 15\right]^T.$$
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