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: 55


$$[x]_S=\left[ \begin {array}{ccc} \frac{3}{4}\\ \frac{1}{4}\end {array} \right].$$

Work Step by Step

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