I already know how to find the matrix of reflection over the $x$-axis (in the plane) (ON-base). By looking at what happens to the standard basis vectors $\hat{e}_1 = (1,0)$ and $\hat{e}_2 = (0,1)$. The matrix for the transformation will be:
$$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$
That's an easy example. But I wan't to find the matrix for the reflection over a line that goes through the origin and makes the angle $\pi/17$ with the x-axis.
How can I approach this problem?
Best Answer
The same way. Look at the standard basis and find out the images of the standard basis under your transformation. These should involve sines and cosines of $\pi/17$...
UPDATE
OK, let's reflect $(1,0)$ over that line. Notice in a sense you would be just rotating the point by $2\pi/17$ radians around the origin, can you figure out the result?
Similarly, from $(0,1)$ to the line is $\pi/2 - \pi/17 = 15\pi/34$ radians, so the result after reflection is a negative rotation of $2 \cdot 15\pi/34$ radians from the $\pi/2$ angle, resulting in the final angle of$$\frac\pi2 - 2 \cdot \frac{15\pi}{34} = \frac{-13\pi}{34}.$$
Can you take it from here?