Question

, what is
?

, which is a reflection over the
-axis.

, which is a reflection over the
-axis.

, which is a reflection over the
-axis.

, which is a reflection over the
-axis.

Answer 1

I don't see the symbols in your message — they were lost. Could you resend the exact composition you want (e.g. R_x ∘ R_y, or R_{y=x} ∘ R_x, etc.)?

While you retype it, here are the usual reflection maps and how to compute compositions (so you can match to your problem):

Common reflections (as maps (x,y) → ... and as 2×2 matrices)
- R_x: reflection across the x-axis: (x,y) ↦ (x, −y). Matrix: [[1,0],[0,−1]].
- R_y: reflection across the y-axis: (x,y) ↦ (−x, y). Matrix: [[−1,0],[0,1]].
- R_{y=x}: reflection across the line y = x: (x,y) ↦ (y,x). Matrix: [[0,1],[1,0]].
- R_{y=-x}: reflection across the line y = −x: (x,y) ↦ (−y, −x). Matrix: [[0,−1],[−1,0]].

How to get a composition
- Represent each reflection by its matrix, then multiply matrices in the correct order. If you want f ∘ g (do g first, then f), compute M_f · M_g.
- Reflections are orthogonal matrices with determinant −1, and the product of two reflections through lines through the origin is a rotation by twice the angle from the second line to the first.

Some useful examples
- R_x ∘ R_y = R_y ∘ R_x = [[−1,0],[0,−1]] = −I (rotation by 180°).
- R_{y=x} ∘ R_x = [[0,−1],[1,0]] (rotation by +90°).
- R_{y=-x} ∘ R_x = [[0,1],[−1,0]] (rotation by −90°).
- R_{y=x} ∘ R_y = [[0,1],[−1,0]] (rotation by −90°).

If you paste the exact expression you meant, I’ll compute that specific composition and give the resulting map (in coordinate formula and matrix form).