Question

Consider a subset alpha={x+x2,1+x2,1 2x+2x2}ofP2(R). (a) Show that alpha is a basis for P2(R). (b) For f(x) = 1 + x + x2 2 P2(R), find its coordinator vector [f] alpha with respect to alpha. (c) Let = {1, x, x2} be the standard basis for P2(R), and let f(x) = a + bx + cx2 and g(x) = p+qx+rx2 be the elements of P2(R) and k 2 R. Prove that [f+g] = [f] +[g] and [kf] = k[f] and conclude that the map T : P2(R) ! R3 defined by T(f) = [f] for f 2 P2(R), is linear.

Consider a subset x= 4x+2, 1+x? 1-2x + 2x²} of P2(R? (o show that a is a basis for P₂ (R) i) For f(x) = 14x48²€ P2 (R), find its a coordinator vector Efla with respect to a. (ii) Letß = {1,x,x²z be the standard basis for P2 (R), and let f(0) = a +bx+ca² and g(x) = p + qx trx ² be the elements of P2 (R) and KER. Prove that If + g Jp = [f]3+ [9] B and [k f] B = K[f] and conclude that the map T = P2 (R) R² defined by Tif)=[f]g for fEP2 (R), is linear.

Answer 1

º d = { a+a", ita, 1-2a + 2** } E P(R) let G, C, C3 are scalars in IR site | © (x + 3) + , (+ ) + . ( - 21 + 2) = 0 >(+03) + x. (6,- 203) + ma lei + C2 +263) = 0 a. CatC3 = 0; e, - 2 C₂ = 0 + (₂ + 26₂=0. C3 = - = - 22 . P(R) Hence, a is let, a +bx+ ex Then a+bat ca linearly independent subset of È P₂ (IR).. lamint a alt 2n+20) + frath) = (26-1-20). (a+20) + (6740-0).(142) +(C-5-e) (12) ☺ ... a spams PL (IR) Hence & is a basis of P₂ (IR) .. . f(a) = 1 + x + À EP (IR) let itxu = e. (x+x) + G. (n+1) + 3 (1+2x+222) i ca + C₂ = 1 &,- 20 = 1 e, +6₂ +263 = 1 → Cz=1-42 = 4; = 1+2(1-() = 3 - 202

+ C + 263 1 3-2, + 6 + 2 (1-(2) = 1 3 = 1 > → - e, + 2 - 2 C2 - 3 = - 4 sesa 3 - 262 = 3-3 = 1 the coordinate [f] Wort 2 is Hence f (x) = atbx + cx EP₂ (R). 96) = p + q* + P(R) B !, m} . Now, (+9) () = (a + b) + (6+9)+ (C++)* & [ftg] = (ate, btq , .c+r) Hm) + f(x) = (a +b) + (b+q) x + ((+82) [f]g + [a]= (aib, c) + (p, q, r) = (att, btq, car) = [f+9] . Heure. [f+= [+]+ [310 — TARR 12 defined Kf(n) aka+kbntken? Tkf] = (ka , kb , ke) =k.(a,b,c) - [kk]= k.[f]_@ =k. [4 IR IR? defined T (f) = [f] B & fc P₂ (IR). Now, 7(879) - [4+9]- [f]g + [a]e (800) - T(4) + T() Vt, ge P,(e) 8 T(CH) = [cfl & c. [f]g (Bez ) - c. T (t) t ffP2 (R) of all scalar CFR & T is linear.