Question

4) The linear transformation L defined by L(p(x)) = p(x)+p(0) maps Pinto P. a) Find the matrix representation of L with respect to the ordered bases l_r"} and {1, 1-x). b) For the vector, p(x) = 2x' +1-2 () find the coordinates of L(p(x)) with respect to the ordered basis{1, 1-x), using the matrix you found in a). Remember to use the coordinate vector of p(x) with respect to the basis {1x2). (ii) Show that they are the weights that work by writing the linear combination with the basis elements and comparing the resulting polynomial to L(p(x)).

Answer 1

(4) L: P3 Pe given by f( b (əs) = $'(x) + P10) L(o L (2) la:) = 0+1 = 1 = 1:1 + 0.(1-2) = 1+0= &L El.1 + 0. (1-) = 22 + 0 2.1 +(-2) (1-3) - L (2²) po 2 [4] = 1 0 - 2 a 2 ta):1 + 1.4 +2.3 (b) (ėb(x) = ax +x-2 [b(xs] = 2 1 2 L (6 (21) 4*+1 -2 42-1. +3 1 2 2.3133 -1 -4 0 - 2 [4] [be] =V. .. [L(play) = 13 3:1 +(-4) (1-2) les) f (b(x)) = 4x-1 [L(P(x)] [ १ ]