Answer
$$[x]_S=\left[ \begin {array}{ccc} \frac{3}{4}\\ \frac{1}{4}\end {array} \right].$$
Work Step by Step
Since we have
$$x= \frac{1}{2} \left(\frac{1}{2},\frac{1}{2}\right)+ \frac{1}{2}(1,0)=\left(\frac{3}{4},\frac{1}{4}\right).$$
then we can write $x$ relative to the standard basis of $R^2$ as follows
$$x= \left(\frac{3}{4},\frac{1}{4}\right)=\frac{3}{4}(1,0)+\frac{1}{4}(0,1) .$$
Thus, the coordinates matrix of $x$ in $R^2$ relative to the
standard basis is
$$[x]_S=\left[ \begin {array}{ccc} \frac{3}{4}\\ \frac{1}{4}\end {array} \right].$$
