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