Question

Problem 3. For e 0,7), let Le CRP be the line that makes angle with the positive c-axis. Let To: R2 → R2 be the reflection about Le. Let Ao e R2,2 be the matrix representation of To. a) Find A0, A1/2, and Af/4. For each answer, justify by drawing the images of ēj = (1,0) and ē, = (0,1) under the corresponding reflection. b) Find an orthonormal basis B = (v1, U2) such that To = CT1/ 6C Justify your answer by drawing the images of ēj = (1,0) and ēz = (0,1) under ce T1/ 6C, and C Tx/ 6C c) Is An diagonalizable for all 8 € (0.)? Explain why or why not.

Answer 1

ANSWER

7 (coso, sino) ER2 sit. To (coso, sinol = (coso, sino) = V. and Tolcosco+ III, sin (0+1)=TO (-sino, coso) = (sino, -coso)= va B=20,=ccoso, sino) I 12=(-sino, coso) I is an on b for R² {=de, ea} Now c&=(coso sind sino coso coso Coso Ao = {TOJE COS = cp {To ] CB - Coso-sino 50 sina sino-coso sino caso

Sin2o roos20 I sin2o – Gos20 ( finding Ao Alla and Atla 0 -1 = to mo find the angle 'o' to= ATICA A. dv., dz} define above > A =+), el cod= ANG che = A6 C6 () = A Il Coose-sino)

Auce for a seconde) CO50-B sine. 2.050- _ gino Basino 1050 of sino coso - walk sing coso at 13 C050 - Lsino ssina - 6B coso & Heal to has a eigen values & and -1 Ag is diagon a lzable. since to has 2 eigen valves &- 1 to is diagonalizable for all o ELO, IT)