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: 64


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

Work Step by Step

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