Question

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

Answer 1

ANSWER:

Given, V= P2 (R) with <F,9> = f ( t ) g(t) dt, T:v>v be the linear operator defined by 100) = x+'() +24(x) +1, ij compute T* (1 +22+ x2). first we determined the matriae representation of I wer.to standard basis of P2 (R) Now, the standard basis of Bir) is {1,X,X2} T(1) * X.0+2.1.+1=3 T(X) = 2.1 + 2X+1 = 33.71 T(X).X.2x + 2-32+1 - 4x? +1 3 I 1 so, [T] 0 3 0 O 0 ។ characteristic polynomial of g is Xr1t) - (4:3)*(1-4) sa, 3 is an eigenvalue of T of multiplicity 2 Now, dim Ez 3-rank CCT]-313) 1 1 3-rank 0 O o 0 1

=> 3-2 = 1 2 value 3 is 2 So, there is only one linearly independent eigen vector Corresponding to the eigen value 3 Nowo, alge braic multiplicity of the elgen and geometric multiplicity of the eigen value 3 isi So, I is not diagonalizable since matrix representations of T are similar wir to different bases and similar matrices have Same characteristic polynomials and we see that Wirito standard. basic of BCR) Tifo diagonalizable , so, there does not exist any onthonormal basis of eigenu abors ß for which (T)e is diagonal. ) Hence, is determined.