Question

16. Let T:P4 → P3 be a transformation defined by T(f(x)) = f'(x). This transformation is A) linear and 1-to-1 B) linear and onto C) not linear D) an isomorphism

Answer 1

let T: P4 → P3 be a transformation defined by, T( f(x)) = f'(x) Let f(x), g(x) E P4 : T(«f(x) + Mg(0)) = («f(x) + f g(x))' sq f'(2) + B grą) Ed T (f(x)) + BT (400) where, Q and B ore scalar Thus, Tis linear. consider, Tlf@)) = = T( g(2) f (x) = g(x) integrating on both both sides c fra) = g(x) + c ; one Thus, f(x) & g(x) T is not one too linear transformation 31, 2, 3, 5, bay are the basis of Pu and & 1,0, xc?, x²y are the basis of B, Tif(0) f'(oC) :: T(=0, TOOC)=1, I cae?) – 230, TO22) -32? TO) 5 433 Thus, range of T = P₂ = T is onto . Tis Jinear and onto, so, ophon B) is correct