I. 2. Write the matrix for reflection in line OA (Class)
Thus line of reflection pass through the origin the transformation is linear and could be expressed by 2 x 2 matrix.
1. Reflect base vectors e1, e2.
2. Images e1' and e2' are columns of the matrix of linear transformation.
3. Check the representation by means of the picture "lion".
Solution:
M={{0.8, 0.6},{0.6,-0.8}}
ApplyMatrix(M, lev)
Question 1
Write the determinant of reflection in line
Question 2
Find the fixed points FP and fixed directions FD
Equation for fixed points: X = MX, i.e.
0.8x + 0.6y = x ⇒ −0.2x + 0.6y = 0
0.6x − 0.8y = y ⇒ 0.6x − 1.8y = 0
gives one linear property x = 3y for mirroring line OA.
Equation for fixed directions (eigenvectors): Mv = λv, i.e.
0.8x + 0.6y = λx ⇒ (0.8 − λ)x + 0.6y = 0
0.6x − 0.8y = λy ⇒ 0.6x − (0.8 + λ)y = 0
Dependent equations yield nontrivial solution: det(M − λE) = 0.
det(M−λE) = − 0.82 + λ2 − 0.62
Zero determinant for λ2 = 1.
λ = 1 yields direction x = 3y for mirroring line OA and λ = -1 yields perpendicular direction.