Question

8. Find the standard matrix representation for each linear operator L: R2 + R2 described below: (a) L rotates each vector 7 by 45° in the clockwise direction. (b) L reflects each vector 7 about the 21 axis and then rotates it 90° in the counterclockwise direction.

Answer 1

Ans:8) L: R2 R² L rotates vector in 450 close wise direction Q e = – 45° general notation transform is given by TEET[,,X,]= X, (050-5*, Sone , Xi Simb +x,cose) =(x, cos(45°) - X, Sun 1-45°), K, s. 1-45°) +*2/C051-459) T(x,x) ly XI va + X H₂ 3 b) L reflects each vector about the X, axis than rotates it goo in the counter closewise direction first for reflection about x, axis then 0 =0° Treflection = (X. (0520 +2₂ 5120, K, Sen 20-X2 Coszel = (x, cos 0° + X₂ In0° , X, Sino - X₂ (050) : (21 , - 82) Now • notates for you Trotation (X,,-X₂) = (x, cose+ X₂ Sino ,X, Sno-X Cose) = (x, cos go+X, sin goo, X, Sing6x₂ cosgol (221x1)

C) first L doubles the leangth 2 I, (X, X ₂) = Leagth of (x,, x2 ) is - (x,-0)+ 1X₂-032 = √x , ? + x² then after doubling leangth X = 202 x = 2x, In Ti (X, X) = (2x4,2x2) Now rotates it 30° counter close wise G=30° T(X., X₂ ) = (x,cose -X₂ Eno, K, Sin OtX, cose) T₂ GXp, 242 ) = (2x 2X, (05/30 - 2X₂ Sin (30°) 2X, Son (30°) +2 ₂ X (os (30°) (2x, x 3 - 13 - 2X2 X 7 , 2x, x 1 + x 2 x 53 ) = (53ki - X2, X, +3X2) L = T2 Ti L (X,X2) = (53x, -X2, x, +53X2) dj first I reflects 7 about line X2=X. So o = 450 45°

Ti(x,x) = 2X, Cos2O + X₂ Ein20 , X, Sn 20 - X₂ Cos2el =(x, Cos (90°) + X₂ Sin (90°) , X, Son (90°) - X2 Cos 1907) (X2, X.) Now projection onto the x, axis Tz (X2,81) = (x2,0) L (X1, X2 ) = (x2,0) L=To Ti
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