## Elementary Linear Algebra 7th Edition

Published by Cengage Learning

# Chapter 4 - Vector Spaces - 4.1 Vectors in Rn - 4.1 Exercises - Page 153: 41

#### Answer

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

#### Work Step by Step

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

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