Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 4 - Vector Spaces - Review Exercises - Page 222: 63


$$[x]_{B'}=\left[ \begin {array}{ccc} 3\\ 1\\0\\1 \end {array} \right].$$

Work Step by Step

Writing $x$ as a linear combination of the basis $B'$ as follows $$x=\left(21, -5, 43, 14\right)= a\left(9, -3, 15, 4\right)+b\left(-3, 0, 0,-1\right)+c\left(0, -5, 6, 8\right)+d(-3, 4,-2, 3).$$ We get the system \begin{align*} 9a-3b-3d&=21\\ -3a-5c+4d&=-5\\ 15a+6c-2d&=43\\ 4a-b+8c+3d&=14. \end{align*} By solving the above system we have the soluiton $$a=3, \quad b= 1, \quad c=0, \quad d=1.$$ Thus, the coordinate matrix of $x$ in $R^3$ relative to the basis $B'$ is $$[x]_{B'}=\left[ \begin {array}{ccc} 3\\ 1\\0\\1 \end {array} \right].$$
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.