Let $f:\Bbb{R}^3 \rightarrow \Bbb{R}^3$, where $f(x_{1},x_{2},x_{3})$ is rotation of $(x_{1},x_{2},x_{3})$ by an angle of $\frac{\pi}{2}$ in the $x_{1}x_{3}$ plane followed by a reflection in the $x_{1}x_{2}$ plane, followed by a stretch in the $x_{3}$ direction by a factor of $5$.
I see them as compositon fo linear transformations undergoing rotation $(T_{1})$ then reflection $(T_{2})$ then stretching $(T_{3})$.
So rotation in the $x_{1}x_{3}$ plane, means that we are rotating around $x_{2}$ axis, that is we are considering the rotation matrix
$\begin{bmatrix} \cos(\theta) & 0 & \sin(\theta) \\ 0 & 1 & 0\\ -\sin(\theta) & 0 & \cos(\theta) \end{bmatrix}$.
So for the rotation angle of $\frac{\pi}{2}$, we have the rotation matrix as $A_{1} = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0\\ -1 & 0 & 0 \end{bmatrix}$ so, $T_{1}(x_{1},x_{2},x_{3}) = (x_{3},x_{2},-x_{1})$.
Next reflection in $x_{1}x_{2}$ plane -
$T_{2} (x_{3},x_{2},-x_{1}) = (x_{3},x_{2},x_{1})$.
Next stretch in the $x_{3}$ direction by a factor of $5$ -
$T_{3}(x_{3},x_{2},x_{1}) = (x_{3},x_{2},5x_{1})$.
So,
$T_3\circ T_2\circ T_1(x_1,x_2,x_3)=(x_3,x_2,5x_1)$
Is my work correct, any ideas to improve?
If $A_{1} = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0\\ -1 & 0 & 0 \end{bmatrix}$ then , $T_{1}(x_{1},x_{2},x_{3}) =\begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0\\ -1 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} x_{3}\\x_{2}\\\color{blue}-x_{1} \end{bmatrix}$.