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: 14


$$[x]_{B'}=\left[ \begin {array}{ccc} 2\\ -4\\3 \end {array} \right].$$

Work Step by Step

Writing $x$ as a linear combination of the basis $B'$ as follows $$x=\left(3,-\frac{1}{2},8\right)= a\left(\frac{3}{2},4,1\right)+b\left(\frac{3}{4},\frac{5}{2},0\right)+c\left(1,\frac{1}{2},2\right).$$ We get the system \begin{align*} 8a+7b+c&=3\\ 11a+4c&=19\\ 10b++6c&=2. \end{align*} By solving the above system we have the soluiton $$a=2, \quad b=-4, \quad c=3.$$ Thus, the coordinate matrix of $x$ in $R^3$ relative to the basis $B'$ is $$[x]_{B'}=\left[ \begin {array}{ccc} 2\\ -4\\3 \end {array} \right].$$
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