Question

4) The linear transformation L defined by L(p(x)) = p'(x)+ p(0) maps P, into P. a) Find the matrix representation of L with respect to the ordered bases {1xx.x"} and {1, 1-x}

Answer 1

Given: The linear transformation is, L(P(x))= p'(x)+ p(0) maps pz to P2. a) The matrix representation of the above linear transform is computed as follows, L(1)=1=1(1)+0(1-x) L(x)=1=1(1)+0(1 – x) 1(x2)= 2x= 2(1) – 2(1-x) Therefore, the matrix can be written as, 1 1 2 0 0 -2

b) The coordinates of L(P(x)), where p(x)=2x²+x-2. The polynomial p(x) can be written as, P(x)=-2(1)+1(x)+2(x²) Now, compute (P(x)) using the matrix in part a), 1 2 3 = [ 0 0 -2 -4 2 Therefore, the coordinates of L(P(x)) is (3,-4), that is L(P(x)) = 4x-1.

c) Consider the second-degree polynomial p(x) = ax +bx+c. L(P(x)) = 2 ax +b+c Represent the above polynomial into basis elements of P2. 2ax +b+c=d(1)+ e(1-x) = d+e-ex Equate the coefficients of x and constant terms of the above equation, d+e=b+c 2a=-e The weights corresponding to the polynomial L(p(x)) is (b+c-e, -2a). Verification of L(P(x)), where p(x)=2x² +x-2. Here, a = 2, b=1 and c=-2. Therefore the coefficients of L(P(x)) is, (3,-4), with respect to the basis {1,1–x}.