Question

Problem 2 A Use Gram-Schmidt orthogonalization to modify {1, 2,2} into an orthonor- mal basis of P, on the interval [0, 1]. B) Note that the operation is linear on P and maps P. R. Hence, find a polynomial qe as in the Representation Theorem such that for any pe P, p" = lo p(x)q(r)da. C) Observe that the operation T linear. Find its adjoint T : P P P P given by Tp = rºp" + Ip' + p is

Answer 1

solution & . Given that @ we have – (say) 3 =449,4"} 7 r Gram Schmidt Orthogonalization for bases B = {v, V2 -- Unys gives orthogonalization Basis B = {4, A2 - Ungis where - k-l var Turun For the equation = .</21 U1=1 U₂ zx a l ot = 1-1/2 1/2-1/2 <liy fiskalis af 1.104

Thus, Us=기를 | | 그i 세리가일 (1 ) NOW) -> <4 Jial) - ( - 32 CA 1 - -- Strid - ( + - )

- Ut3-6 12 to 43 -4° 144) 43 = x2 - - +4+3 = 6924+4 Let's Ortho normalizatoon U, U2, 43 as - 4'- - Uz 43 lugh!

98 Ohis - 21-1421 + $ 81 - 1+ 1 224 7+ la - Sem4***-*** 3 terjay's *. tuna** *4jag? han (4) SIP, CH), Joy oh l * Tik 'ni'n> )= min il * uant MON

(3) (toba) Vie Basis of fz in So, orthomornal interval Cag) or { "JiB (x++), 615 (224+5)

Let axa di (PCA)) =< PC ay lack) - 0 . Where q(X) = atbx tcare so, from eqh 6 one (1){ 1/2(2)>= a + +9 0 or Ga + 3b+2c=0 -☺ de (n) = (x1204) = daa 0 = a + b a te Ga+ 46+30=0 —

Thos d2 a (x²= <x/q (x) = 2 or as to y te g = 2 - from equation ® or@ we get bte=o -b= +c]—© so, from © by putting C=-6 weget . from @ $(-) + y b = 2 - hot --- b: - 180x2

=> b=-360 C = 360 a: 60 12cm) - 60*3604 +3604 As we have orthonormal Basis B={ 1, J12 (x-1),655 (x²x+1 )} of I capo ba (x) Lazca) P2. in [i]. Let's fond matrux of T with respect to thies Orthonormal Bases *TaillT(1) = x(0"+x (I)'+1 = 1 = 1.9,0) 40. a, (ale) ..to azlas

*T (92(x) = x²0ta (12) + J12 (a = 5124 +12 (2-1/2) = 25JTB (x-1) + VIZ QMX) + 2162 ) +0.93 (1) * T Caz(x) =*? (655)+*(655 (23-1) +6512xy = 655x+12/54-651 +61819- 655*+55 & 2455 x ²-1255x+55 = y az (0) + 12550 -315 Ey 9319)+ 560,4,693 +355

Thos = 355 a, (x) + 560 02 (2) + U 93(a) C J12/2 355 1 21 7777 B 375 Too 4 Albis Ortho normal Basos.