Answer
$\mathbf{v}=0\mathbf{u}+\mathbf{w}$
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 $\mathbf{v}=\mathbf{w}$
