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: 11


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

Work Step by Step

Writing $x$ as a linear combination of the basis $B'$ as follows $$x=(12,6)= a(4,0)+b(0,3).$$ We get the system \begin{align*} 4a&=12\\ 3b&=6. \end{align*} By solving the above system we have the soluiton $$a=3, \quad b=2.$$ Thus, the coordinate matrix of $x$ in $R^2$ relative to the basis $B'$ is $$[x]_{B'}=\left[ \begin {array}{ccc} 3\\ 2 \end {array} \right].$$
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