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: 15


$$[x]_{B'}=\left[ \begin {array}{ccc} 0\\ -1\\2 \end {array} \right].$$

Work Step by Step

Writing $x$ as a linear combination of the basis $B'$ as follows $$x=\left(11,18,-7\right)= a\left(4,3,3\right)+b\left(-11,0,11\right)+c\left(0,9,2\right).$$ We get the system \begin{align*} 4a-11b&=11\\ 3a+9c&=18\\ 3a+11b+2c&=-7. \end{align*} By solving the above system we have the soluiton $$a=0, \quad b=-1, \quad c=2.$$ Thus, the coordinate matrix of $x$ in $R^3$ relative to the basis $B'$ is $$[x]_{B'}=\left[ \begin {array}{ccc} 0\\ -1\\2 \end {array} \right].$$
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