How are the transformation matrices obtained for the operations like $C_2, C_2^{'}, C_2^{''}$ in this case 
, where the results of application of symmetry operations are given as: 
and the unitrary representations are given in the book like 
Though I understand the unitrary representations of $I, C_3, C_3^{2}$ from the concept of rotation of body set of axes relative to an external space set of axes in 2D plane, I find it inapplicable with the given red marked results.
Further, in the case of $U(C_2^{'})$, if we apploach like $\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right) \stackrel{U(C_2^{'})}{\rightarrow}(0,1)$
$$\left[\begin{array}{c} \frac{\sqrt{3}}{2} \\ -\frac{1}{2} \end{array}\right]=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\left[\begin{array}{l} 0 \\ 1 \end{array}\right]$$, then we find that $U(C_2^{'})=\left[\begin{array}{cc} +\frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{array}\right]$ is suitable for the sake of Matrix Multiplication result, rather than $\left[\begin{array}{cc} -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & +\frac{1}{2} \end{array}\right]$ as given in this book. Though I still don't know how to obtain $U(C_2^{'})$ for given $\left[\begin{array}{c} \frac{\sqrt{3}}{2} \\ -\frac{1}{2} \end{array}\right] \&\left[\begin{array}{l} 0 \\ 1 \end{array}\right]$. Can anybody please explain this to me? Thank you in advance.