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 - 4.1 Vectors in Rn - 4.1 Exercises - Page 153: 45


$$v= u_1+2u_2-3u_3.$$

Work Step by Step

Suppose the linear combination $v= au_1+bu_2+cu_3$, then we have $$(10,1,4)=a(2,3,5)+b(-1,2,4)+c(-2,2,3), \quad a,b,c\in R.$$ Which yields the following system of equations \begin{align*} 2a-b-2c&=10\\ 3a+2b+2c&=1\\ 4a+4b+3c&=4. \end{align*} Solving the above system, we get the solution $a=1$, $b=2$, $c=-3$. Hence, $$v= u_1+2u_2-3u_3.$$
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