# 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.