Question

Problem 13. (7 points) Determine which of the following transformations are linear transformations 1. The transformation T defined by T2, 13, 13) = (21,2,3).? 2. The transformation T defined by T (21,22) = (x1,20;22). ? 3. The transformation T defined by T(31,42) = (4.01 - 222,312) ? " 4. The transformation T defined by T (21,12) = (2x - 3x2,41 +4,221). ? 5. The transformation T defined by T (11, 12, 13) = (0,0,0). ? Note: You can earn partial credit on this problem Type here to search

Answer 1

Problem 13 Hore T is defined by T(A4, z, y) = (49 2,3) et x=(2, 22, 4) and y = (41, 42, 3) & TR² Then x+y = (2+ Geg 2 + y 2, 8z +43) Now +(x+4)= (1 + Ye + 2q +42,3) T(X) = (x13 m2, 3), 7(n)= (.14283) T(X) + T(Y)= (21+ 4 xq +4206) TEN+TY) & TEXTO) for xgy E IR² Hence T is not a linear trous formation. Here is defined by a T(Mod) = (x4, x. x) rex (24, 2) and (y. Y) ER2 NOO T (14, N2) = (al, x s. 8) T(4) T (41942) - (429 41.42) T(x+4)=+(4+4 , 82 +42) = (2+ Yen (4+41)(*2742)) T +T() = (x4 + 4 + 4x +4192) since (+4) (2 + y2) & run +442 so t (x+4) & + (x) + TC4, foro xay ER2 Hence T ý not a line ar transform

. Here T is defined by T(3493)= (434- 2x2 ,332) .net x = (24, 22) and y= (41, G₂) € 1R ² Nad T(X)= (4%- 2xy, 3%) T(Y) = (44, 242, 342) +(«) + T(y) = ( 4 84 +441-232 – 242,3x2+ 342) T(+y) = T(24+ 41, 92442) ( 4 (9474) - 2 (342742), 3 (974) .( 4 24+ 4y - 232-272 232+342) get CER No T(CX) = T (ex, exn] P C 4224-2exy, 30%) że ( 424 - 223 , 3X2) = CRT(X) This +(x+4)= T(X) + T(Y) and +(6x) = CT(X), & RYE IR² and a Hure tis a linear transformation. 4. Here t s defined by T(2, 12) - (224 – 34224+421931) ser x=(4, Hz) and y=(4,42) E1R2 Non T(X)= (224 - 3%, 4 + 4 , 1821) +14) = (24-342, Ye +4, 1921) C(x+1) = n( 4 +41 , 2 + ) (2(a + y ) - 3 (M + 42), 24 + 4 + 4 , 1X2441) (224+241- 342-342 , auty, 1-349-3422 42+ +4 ,1%+421)

70+ Tey) = (2x4+24 -322-342 4 + Ye +8 , P3/+1921) This T(x+4) & TEX) + Tey) Hence & is not a line ar for (x,y) & R2 transformation, 5. Here T's defined by s o me T(4332926 ) = (0,0,0) ser n= (24, z, sey) and s= (5424) E IR 3- Then 7(x)= (0,0,0). Tye (0,0,0) TRAY) = T( + G + 42, 3 & 43) . TEXTY) = T(a) + Tey) gt Raye IR3 het CER Than T(ex) = T(exe, e, em FC. (0,0,0) = C TX Thus Tary) = T (U+ Tey) and Tex)=CTON. & X,YER² and CER Therefore t is a Linear transformativo ast