Question

1. Let L: P1(R) + P1(R) be a linear transformation given by L(a + bx) = a - b + (2a – b)x. Let S = {1, 2} and T = {1+x} be two basis for P1(R). (a) Find the matrix A of L with respect to basis S. (a) Find the matrix B of L with respect to basis T. (c) Find the matrix P obtained by expressing vectors in basis T in terms of vectors in basis (d) Verify the change of base formula AP = PB.

Answer 1

I Defined, Llar baj Let (a) Defined, Llar bx) = a-b+ (2a-b2. S=21,8) be the basis of Pi (IR) Now L (1) = 2(1+0) 1 + 2% 1 + 2- - 2(x) = 2 (0+11) +(-1) (-1)) +(-1X Hence e A2 -11 (6) 12-1 Let I-41, 1+x) be another basis of P, OR) 211) = 14+ 2. = (-1) + 2(1+0) 24(1+) = (-1) + ( 2-1-1) 0 it in

L*(1+x) = -1 ++(1+x) - (-1).1 + 1(1+) Henee Hence B = 2 il © 1) = 1+22 Iton - it 2-2 Lli+x) = x = 0.lt . = (0)1 + 1. (*) Hence p= (2) (2) Now AP = (2-1) (2) pe = (0)65)-11-1)

Hence eo AP= PB (verified) Note :- The given sel T-h1+h) is Singleton set and dim (P (IR)) = 2. So I is not a basis of P, (IR). we may assume To 41, 1+2) is a basis of P. 0k).