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: 43



Work Step by Step

Write the vector $\mathbf{v}=(-1,-2)$ as a linear combination of $\mathbf{u}=(1,2)$ and $\mathbf{w}=(1,-1)$ If $\mathbf{v}=a\mathbf{u}+b\mathbf{w}$ Then $-1=a+b$ $-2=2a-b$ If we add these equations together we get $-3=3a\implies a=-1$ Substituting this value back into our first equation gives us $-1=-1+b\implies b=0$ This is consistent with our second equation: $-2=2\times(-1)$ So our linear combination is $\mathbf{v}=-\mathbf{u}+0\mathbf{w}$ We could have alternatively solved this by just recognizing that the $\mathbf{v}=-\mathbf{u}$
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