Answer
$$[x]_{B'}=\left[ \begin {array}{ccc} 2\\\frac{3}{2} \end {array} \right].$$
Work Step by Step
Writing $x$ as a linear combination of the basis $B'$ as follows
$$x=(2,-1)= a(1,1)+b(0,-2).$$
We get the system
\begin{align*}
a&=2\\
a-2b&=-1.
\end{align*}
By solving the above system we have the soluiton
$$a=2, \quad b=\frac{3}{2}.$$
Thus, the coordinate matrix of $x$ in $R^2$ relative to the
basis $B'$ is
$$[x]_{B'}=\left[ \begin {array}{ccc} 2\\\frac{3}{2} \end {array} \right].$$
