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 - 4.7 Coordinates and Change of Basis - 4.7 Exercises - Page 210: 16


$$[x]_{B'}=\left[ \begin {array}{ccc} -1\\ 1\\3\\2 \end {array} \right].$$

Work Step by Step

Writing $x$ as a linear combination of the basis $B'$ as follows $$x=\left(0,-20,7,15\right)= a\left(9,-3,15,4\right)+b\left(3,0,0,1\right)+c\left(0,-5,6,8\right)+d(3,-4,2,-3).$$ We get the system \begin{align*} 9a+3b+3d&=0\\ -3a-5c-4d&=-20\\ 15a+6c+2d&=7\\ 4a+b+8c-3d&=15. \end{align*} By solving the above system we have the soluiton $$a=-1, \quad b= 1, \quad c=3, \quad d=2.$$ Thus, the coordinate matrix of $x$ in $R^3$ relative to the basis $B'$ is $$[x]_{B'}=\left[ \begin {array}{ccc} -1\\ 1\\3\\2 \end {array} \right].$$
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