Mirror the F.
Build the matrix that flips the plane left to right, like a mirror standing on the vertical axis. The F keeps its size. It only faces the other way.
Think: the word on an ambulance
AMBULANCE is printed backwards on the front of the vehicle. In a driver's mirror it reads the right way round. A mirror keeps sizes and distances but reverses left and right.
How to build it
Use the rule from slide 5: decide where the two landmarks land, then write them as the columns.
- A mirror on the vertical axis sends points on the right over to the left.
- Points on the vertical axis itself stay where they are.
What the determinant says
The F keeps its size, so |det A| = 1.
But the F now reads backwards, like turning a sheet of paper over to its other side. No rotation can do that. The determinant signals the flip with a minus sign. Watch the readout under the plane.
After you solve it
- Mirror twice: two mirrors in a barber's shop show your face the right way round. Check that A @ A gives the identity.
- Another mirror: try the mirror on the diagonal line y = x. Where do (1, 0) and (0, 1) go?
Worked examples
Example 1 · Mirror on y = x
(1, 0) → (0, 1) and (0, 1) → (1, 0).
Example 2 · Two mirrors make a turn
Mirror on the y-axis, then on the x-axis:
Two flips cancel. The result is the half turn: no mirror image left.
Problems
Try each one before you open the solution.
Write the mirror across the x-axis. Where does (3, 2) go?
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M = 100−1. (3, 2) → (3, −2).
Check that the mirror on y = x, applied twice, gives the identity.
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0110² = 1001. Swapping the coordinates twice puts them back.
Is 0−1−10 a mirror? Which line?
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det = 0 − 1 = −1 and M² = I, so yes. (1, 0) → (0, −1) and (1, −1) stays put: it is the mirror on the line y = −x.