## Elementary Linear Algebra 7th Edition

$\mathbf{v}=\mathbf{u}+\mathbf{w}$
Write $\mathbf{v}=(2,1)$ as a linear combination of $\mathbf{u}=(1,2)$ and $\mathbf{w}=(1,-1)$ If $\mathbf{v}=a\mathbf{u}+b\mathbf{w}$ Then $2=a+b$ $1=2a-b$ Adding these equations gives us $3=3a\implies a=1$ Plugging this value back into our first equation gives us $2=1+b\implies b=1$ And so our linear combination is simply just the sum $\mathbf{v}=\mathbf{u}+\mathbf{w}$