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: 42



Work Step by Step

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