Question

(12) (after 3.3) (a) Find a linear transformation T. Rº Rº such that T(x) = Ax that reflects a vector (1), 12) about the Tz-axis. (b) Find a linear transformation SR2 R2 such that T(x) = Bx that rotates a vector (2, 2) counterclockwise by 135 degrees. (c) Find a linear transformation (with domain and codomain) that has the effect of first reflecting as in (a) and then rotating as in (b). Give the matrix of this transformation explicitly. How is this transformation related to T and S? (d) Find the matrix representing the linear transformation that first rotates as in (b), then reflects as in (a), and then rotates backwards (i.e., clockwise by 135 degrees). (e) What matrix do you get if you repeat the sequence in part (a) ten times? Write this matrix in terms of A and B. Can you write this matrix explicitly?

Answer 1

(2) TR 1R2 T(x)= Reflection about garis - 1x (10) + T(LO)= (-1,0) Tol) - (01) = 0 (0) (0) 1 A=IT] or loi Si R² 182 ВМС. in anticlock The rotation matrix of an angle o B = Hero o=1350 Sh1350 of 1350 ...B-1516- Thomas - 2 - If T' be the the theor transtornatior. that effect of tft reflection and ther rotating meget IT'] = TSJO[T

T' (open) is o T (A2) - T' (2,4) : SCT (hire) ) Scaxot ) = To coso - sime - sing + cofu :. The required matrix is IT'] = [3] IT] -1.5 208 1350 - sm 1350 Shis Cof 1350 ] ] 110) tz - -] u v2

() 17") - IT]o15] 17"). Toll - cel hed ea ta + / lo i Ii] (iedacixer -10 eso enxena Toi @ 102 Now t - 1/2 0 АП erreed)