Example I If the vectors $\boldsymbol{a}=(1,1), \boldsymbol{b}=(1,-1), \boldsymbol{c}=(-1,2)$, then $\boldsymbol{c}$ equals
A. $-\frac{1}{2} \boldsymbol{a}+\frac{3}{2} \boldsymbol{b}$
B. $\frac{1}{2} a-\frac{3}{2} b$
C. $\frac{3}{2} \boldsymbol{a}-\frac{1}{2} \boldsymbol{b}$
D. $-\frac{3}{2} a+\frac{1}{2} b$
Let $c=m a+n b$, then
$$
(-1,2)=m(1,1)+n(1,-1)=(m+n, m-n)
$$

So $\left\{\begin{array}{l}m+n=-1 \\ m-n=2\end{array}\right.$
Solving, we get $\left\{\begin{array}{l}m=\frac{1}{2} \\ n=-\frac{3}{2}\end{array}\right.$
Therefore, the answer is B.