Question

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

Answer 1

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NOW? I du C C Vē s du =[^]' =1-6-1) = 2 ✓nly <**) = Sandu S'xron D 3 [1-419 1 2 linll. vamins ra 3 = v # 5x, nn) s'aranda [na ] 16-6-15 ] [i+1] 5 5 11 Xull Takwa ya ✓ N3 SAN

0 onthonormal basid One of the required *ial, un G 2 || 1| 1 x 1 R 言, ) VR { 友, EN ys 5 x" .comsiting of eigen vectors .”