Question

2. Consider the inner product space V = P2(R) with (5.9) = £ 5(0)9(e) dt, and let T:V V be the linear operator defined by T(f) = xf'(2) +2f(x). (i) Compute T*(1+2+x²). (ii) Determine whether or not there is an orthonormal basis of eigenvectors 8 for which [T], is diagonal. If such a basis exists, find one.

Answer 1

0 Given that Let, V:B(IR) Tvv be a lhear operatos Tlf): xfYv) +28 (1) Now, a= {1, 2, x} is a basis of v. T1: X.0+2:112=2.17 0.870.4" = 2 T(x)3 2014 2.72 =32 -0.1+3x tourer - 37 T(x)= x-2x+2:x°=4798.0.1 +0.2 +4072 = 4* So the matriu of T is o [TJa 3 o T(say simply o Now, it 2 0 T ☺ o 4 20 0 3 o 4 0 0 so - T* (14 px + x² ) = T (1+x+*% - TUU+T+Tlxy = 2+3x +43(banco).

2,3,4 vectors ß for which , [T] B is diagonal The eigen values of T one all eigen values are distinct So, if is diagonaligible over R a Symmetric matrix gonally diagonaligible over an ortho normal basis of eigen matrin (v po and y be since T is So, I is orthog st we observe that be T(1)=2.) 50,1 is eigen vector T(x) = 422 eigen vector TV dv d = eigen value corresponding to 3 So, x is eigen vector corresponding to 4 so, {1,x,xy] is required orthogonal basis of eigen vectors of t

3 Now, 21,17 - 1 Slide Så (-2'= v-bilia illill=eilyar Luix> = x.xdu n LE Sarde = nput - al [i-ki = 12:2 ::))x)= x,x) 5 % 13 Lurian) = f x² x r dr. 5 :ls [ 15.6-18 =% [141] = 2/5 Lu? 12 2. √2

on of the required requéved orthonamal n bais 2 11411 71 B-listan { / Era 1 V2 B V 5 a) } a ra consisting of eigen vectors of T