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: 10


$$[x]_S=\left[ \begin {array}{ccc} -20\\ 32\\-4\\-5\end {array} \right].$$

Work Step by Step

Since we have $$x= -2\left(4,0,7,3\right)+3\left(0,5,-1,-1\right)+4\left(-3,4,2,1\right)+(0,1,5,0)=\left(-20,32,-4,-5\right).$$ then we can write $x$ relative to the standard basis of $R^4$ as follows $$x= \left(-20,32,-4,-5\right)=-20(1,0,0,0)+32 (0,1,0,0)-4(0,0,1,0)-5(0,0,0,1) .$$ Thus, he coordinates matrix of $x$ in $R^4$ relative to the standard basis is $$[x]_S=\left[ \begin {array}{ccc} -20\\ 32\\-4\\-5\end {array} \right].$$
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