Challenge: build a mirror
A challenge that uses slide 1: build the matrix for the mirror across the y-axis (the vertical axis). It swaps left and right. The F should keep its size and just face the other way.
Lesson 6 · slide 2Reflection- To build a matrix, decide where 10 and 01 land and use them as columns (slide 1).
- A mirror keeps lengths and areas, so |det A| = 1.
- A mirror flips the plane over, and the determinant shows that with a minus sign: det A = −1.
- Applying the same mirror twice brings everything back: A·A = I. Here I = 1001 is the identity matrix, which leaves every vector where it is.
A mirror you have seen
AMBULANCE is printed backwards on the front of the vehicle, so drivers read it the right way round in their mirror. A mirror keeps every size and distance, but swaps left and right.
How to build it
Use the rule from slide 1: where should the two landmarks go?
- The mirror sends points on the right over to the left. So 10 goes to −10.
- Points on the y-axis stay put. So 01 stays at 01.
Those are the columns. Write them in, run the code, and watch the F.
What the determinant tells us
The F keeps its size, so |det A| must be 1. But it now reads backwards, like a sheet of paper turned over. No rotation can do that. The determinant marks the flip with a minus sign: det A = −1. Check the readout under the picture.
Keep reading with the full course
Finish lesson 6 and you will be able to: see a matrix as a move of the plane, and build one from where two arrows land.
Still locked here:
- The rest of this explanation
- 4 more examples with step-by-step solutions
- Five extra members-only problems with worked solutions
$20 once, lifetime access. Lessons 1 to 3 stay free.
Examples
Example 1 · Mirror across the line y = x
10 → 01 and 01 → 10.
Questions and comments
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