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: 62


$$[x]_{B'}=\left[ \begin {array}{ccc} -5\\-11\\ 9 \end {array} \right].$$

Work Step by Step

Writing $x$ as a linear combination of the basis $B'$ as follows $$x=(4,-2,9)= a(1,0,0)+b(0,1,0)+c(1,1,1).$$ We get the system \begin{align*} a+c&=4\\ b+ c&=-2\\ c&=9. \end{align*} By solving the above system we have the soluiton $$a=-5, \quad b=-11, \quad c=9.$$ Thus, the coordinate matrix of $x$ in $R^3$ relative to the basis $B'$ is $$[x]_{B'}=\left[ \begin {array}{ccc} -5\\-11\\ 9 \end {array} \right].$$
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