# I. 2. Write the matrix for reflection in line OA (Class)

- Author:
- Šárka Voráčová

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)*=

*−*0.8

^{2}+

*λ*

^{2 }

*−*0.6

^{2}Zero determinant for

*λ*

^{2 }= 1.

*λ*

^{ }= 1 yields direction

*x = 3y*for mirroring line OA and

*λ*

^{ }= -1 yields perpendicular direction.