Is the following statement correct?
If the measure of the angle between lines $\ell$ and $m$ in the plane is $\theta$, the reflection of the plane across both lines is the same as the rotation of the plane by $2\theta$ about the intersection of the two lines.
I would appreciate a reference to an explanation to this statement (or to an analogous statement). An explanation using linear algebra is preferable.
It's correct, yes.
I bring neither linear algebra (although a recent answer of mine does talk about about what makes a rotation matrix in $\Bbb R^2$, so you can just compute things) nor references nor a proof, but instead a picture:
Here we have lines $m$ and $\ell$, separated by angle $\theta$, and two colored points for reference.
We apply the reflection across $\ell$ (call this transformation $r_\ell$). The point on $\ell$ is fixed, and we also keep track of where $r_\ell$ sends the line $m$; the new line we call $r_\ell(m)$. Notice that $m$ and $r_\ell(m)$ are separated by $2\theta$.
A similar thing happens applying $r_m$, and now we keep track of where $r_m$ sends two lines. At any rate, the green point that started on $\ell$ is moved onto the line $r_m(\ell)$, where the lines $\ell$ and $r_m(\ell)$ form an angle of $2\theta.$ The red point is moved from line $m$ to line $r_m(r_\ell(m))$, also separated by an angle of $2\theta$.
It's not at all formal, but hopefully provides some insight.