# Find the matrix of linear transformation (in Standard basis) that rotates clockwise every vector

Find the matrix of linear transformation (in Standard basis) that rotates clockwise every vector in $\mathbb{R}^{2}$ through an angle $\pi/4$ and then reflects it across y axis.  The standard matrix of rotation by $\pi/4 \ clockwise$ is R = $\begin{pmatrix} cos(\pi/4) & \sin (\pi/4) \\ -\sin(\pi/4) & cos(\pi/4) \end{pmatrix}$= $\begin{pmatrix} 1/\sqrt2 & 1/\sqrt 2 \\ -1/\sqrt 2 & 1/\sqrt 2 \end{pmatrix}$ . Now reflection matrix about y axis is R'= $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ . Hence composition of Rotation and Reflection is $\ R \circ R'=\begin{pmatrix} 1/\sqrt 2 & 1/\sqrt 2 \\ -1/\sqrt 2 & 1/\sqrt 2 \end{pmatrix} \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ . But I am not sure , please help me

• Multiplying these matrices one by one with a vector, which operatorn is applied first? Apr 10 '17 at 20:39

The product of two matrices is not commutative. If you want the matrix that represents first the rotation than the reflection, the correct order is $R'\cdot R$.