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Find the matrix for the transformation which first reflects across the main diagonal, then projects onto the line $2y+\sqrt{3}x=0$, and then reflects about the line $\sqrt{3}y=2x$.

Reflection about the line $y=x: T(x,y)=(y,x)$, so the standard matrix for this would be the matrix

$$\begin{pmatrix}0 &1 \\ 1 &0 \end{pmatrix}.$$

However I'm not sure how to deal with equations rather than axis. I assume in the second projection, you can simplify it to $y=\frac{-\sqrt{3}}{2} x$. Can you then separate these into a scalar operation $\frac{-\sqrt{3}}{2}$ and orthogonal operation $y=x$? Even so, I wouldn't know how to go further than this since I only know how to do it among the axis.

Edit: $T_1=\left[\begin{array}{cc}0&1\\1&0\end{array}\right]$ by reflecting across main diagonal

$2y+sqrt{3}x=0\rightarrow y=\frac{-\sqrt{3}}{2}x$

$T_2=\frac{-\sqrt{3}}{2}\left[\begin{array}{cc}\cos-45&-\sin-45\\\sin-45&\cos-45\end{array}\right]\left[\begin{array}{cc}0&0\\0&1\end{array}\right]\left[\begin{array}{cc}\cos45&-\sin45\\\sin45&\cos45\end{array}\right]$ by rotating 45 degrees, projecting along the y axis, then rotating -45 degrees (transformations are applied right to left)

$\sqrt{3}y=2x\rightarrow y=\frac{2}{\sqrt{3}}x$

$T_3=\frac{2}{\sqrt{3}}\left[\begin{array}{cc}0&1\\1&0\end{array}\right]$ by separating fraction and then reflecting along $y=x$

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You got a sign wrong for the equation of the line for the second transformation. –  J. M. Apr 2 '13 at 18:28
Fixed, thanks for catching that. –  Manu Juneja Apr 2 '13 at 18:34
I tried this method and edited the original. Can you please confirm that I did it correctly? I projected onto y axis instead because rotating 45 degrees would put it on the y axis –  Manu Juneja Apr 2 '13 at 19:02
Trying that now, thanks. I edited my answer, I didn't realize that transformations are applied that way. –  Manu Juneja Apr 2 '13 at 19:12
I've added my completed answer. Please let me know if it is correct. Additionally, do you know how I should format the answer? I'm not sure what format to put it in other than separating the three transformations –  Manu Juneja Apr 2 '13 at 19:26
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