Final answer:
The matrix of reflection through the line y=2x is determined using the angle of inclination, resulting in the reflection matrix R with entries cos(2θ), sin(2θ), sin(2θ), and -cos(2θ). For θ = 45 degrees, the specific matrix is [[0, 1], [1, 0]].
Step-by-step explanation:
To find the matrix of reflection through the line y = 2x, we can follow a method of constructing a reflection matrix that uses the angle of inclination of the line, which in this case is 45 degrees since tan(45°) = 2. The reflection matrix R for reflecting across a line with angle θ is given by:
R =
cos(2θ)sin(2θ)
sin(2θ)-cos(2θ)
Substitute θ = 45 degrees (or π/4 radians) into the matrix to find the specific reflection matrix for y=2x:
R =
cos(π/2)sin(π/2)
sin(π/2)-cos(π/2)
Which simplifies to:
R =
01
10
This is the reflection matrix you will use to transform any point to its reflection across the line y=2x.