Find the standard matrix of the transformation $T:\mathbb{R}^2\to \mathbb{R}^2$ that corresponds to the reflection through the line Find the standard matrix of the transformation $T:\mathbb{R}^2\to \mathbb{R}^2$ that corresponds to the reflection through the line $x_2=2x_1$ followed by reflection through the line $x_1=3x_2$
I am very confused about this question, could anyone explain how to solve the problem? THANKS!

First you change coordinates to where your basis is given by $\begin{pmatrix} 1 \\ 2\end{pmatrix}$ and $\begin{pmatrix} 2 \\ -1\end{pmatrix}$.
This is done by multiplying by
$$\begin{pmatrix} 1 & 2 \\ 2 & -1\end{pmatrix}^{-1} = \frac{1}{5}\begin{pmatrix} 1 & 2 \\ 2 & -1\end{pmatrix}.$$
Next, you reflect the second coordinate. This is done by multiplying by
$$ \begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}.$$
Finally, you change your coordinates back by multiplying by
$$\begin{pmatrix} 1 & 2 \\ 2 & -1\end{pmatrix}.$$
Thus, your transformation is multiplication by
$$ \frac{1}{5}\begin{pmatrix} 1 & 2 \\ 2 & -1\end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}\begin{pmatrix} 1 & 2 \\ 2 & -1\end{pmatrix} = \frac{1}{5}\begin{pmatrix} 1 & 2 \\ 2 & -1\end{pmatrix}\begin{pmatrix} 1 & 2 \\ -2 & 1\end{pmatrix} = \frac{1}{5}\begin{pmatrix}-3 & 4 \\ 4 & 3 \end{pmatrix}.$$
Now try to do the second reflection yourself.
