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I am given $T:\mathbb{R}^2\to \mathbb{R}^2$ a linear map that rotates the points throuh $-2\pi/3$ and then reflects the points through vertical $y$ axis. So basically $T=T_1\circ T_2$, where $T_2$ is rotation and $T_1$ is reflection.

I know the rotation matrix is $\begin{pmatrix}\cos\theta & -\sin\theta\\\sin\theta &\cos \theta\end{pmatrix}$

so when $\theta =-2pi/3$ I will get $T_2$ matrix , now how will I get the final answer? Thanks

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The matrix corresponding to $T_1$ is $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}.$

You just have to compute $$\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} $$

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I think you are stuck in identifying $T_1$ for which you should consider just vectors linearly independent vectors $(1,0)$ and $(0,1)$ in $\mathbb R^2$ and their images after reflecting through $y-$axis.

$T_1(1,0)=(-1,0)=-1.(1,0)+0.(0,1)$ and $T_1(0,1)=(0,1)=0.(1,0)+1.(0,1)$


Hence, $T_1=$$$\begin{pmatrix}-1&0 \\ 0&1\\\end{pmatrix}$$

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