Question

Exercise 2 Let B= (Po, P1, P2) be the standard basis for P2 and B= (91,92,93) where: 91 = 1+2,92 = x+r2 and 43 = 2 + x + x2 1. Show that S is a basis for P2. 2. Find the transition matrix PsB 3. Find the transition matrix PB-5 4. Let u=3+ 2.c + 2.ra. Deduce the coordinate vector for u relative to S.

Answer 1

(1) S = (9,,9 2,23) - q = 1+2 q=x+2² 2 3 = 2 + 2+26² Let aq, + bq2+Cq3=0 (1+2) a +(2+2²) b + (2+2+28) C=0 comparing the cofficient a+2c= 0 ) a=-20 a + b + c= 0 b+c= 0 = b =-c from -2c-ct cao

a=0 and b=0 lence q, 22, 23 are linearly independent Since dim P2 = 3 and {2,2, 23} is L-T Subset of 2 80 84,22,23 is a basis of P2 2 let Po=1= aq, + bq + Caz I= a (1+x) + b (x+22²) + ((2+2 +22 comparing the cofficient

I 9+2 c follo o = a + b + c o late > b=-C a=0 C = 1/2 b=-1/2 and similar P = = edit Faq & J 23 = a (174) + f ( x + ²) + g (2 +4 +2

& + 2g = 0 efftgo ح f-1 1 ا

6 = 72 = h (n+1) +i (4+ n) + f (over 1 m) O ht2q -0. 0= ht it & - 1 - i+g - i= 1- f put in h+ 1-ftg = 0 That Transition natix SYB = 10 -1/2 + 1/2

(11) q = 1+0=1+2+04² 22 = x + x² = 0.1 +2 +22 2 = 2+x+x² = 2.1 + x + x² 181 1 (1) u=3+2x+2 ze² = aq + 6% + CV = a (1+x) + b(x + x²) + c(2+x+x²)

comparing - ... a + 20 = 3 a+b +c =2 btc=2 ... a = o C = 36 b = 1/2 :: Coordinati vector = (a,b,c) = co, 1, 32)