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: 44

Answer

$\mathbf{v}=-\mathbf{u}+2\mathbf{w}$

Work Step by Step

Write the vector $\mathbf{v}=(1,-4)$ as a linear combination of $\mathbf{u}=(1,2)$ and $\mathbf{w}=(1,-1)$ If $\mathbf{v}=a\mathbf{u}+b\mathbf{w}$ Then $1=a+b$ $-4=2a-b$ If we add these equations together we get $-3=3a\implies a=-1$ Substituting this value back into our first equation gives us $1=-1+b\implies b=2$ This is consistent with our second equation: $-4=2\times(-1)-2$ So our linear combination is $\mathbf{v}=-\mathbf{u}+2\mathbf{w}$
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