Question

2. Consider the inner product space V = P2(R) with (5,9) = L5(0956 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++r). (ii) Determine whether or not there is an orthonormal basis of eigenvectors B for which [T]3 is diagonal. If such a basis exists, find one.

Please do Part (ii). Feel free to do Part (i).

Answer 1

(ii) Here we first form the transformation matrix T.

We then check if it is diagonalisable. If it is diagonalisable, then the set of Eigen vectors corresponding to its Eigen values forms the orthogonal basis B such that [ T_​​B] is diagonal matrix.

Note : A diagonal matrix is always diagonalisable and the columns of the matrix represent the corresponding Eigen vectors to the diagonal entry as Eigen value.

We divide the vectors by their magnitude to form orthonormal vectors.

2 cis TJ = xc f'(x) + 2 f(e) 6. T(1) = x (o) + 2(1) 2 tox tox osc? T(XC) sc (1) + 2() 0 + 3x + 0x² T(x2) x (2 sc) + 2(x?) = 0+ 0x tux² . T = 2 diagonal matrix 0 3 CS Scanned with CamScanner 니

Now eigen vectors for : y = 2 O d=3 ( 3 ) = 4 - =) form orthogonal basis. { (3) (0) (} form ortogenal basis =) =) we divide them by their magnitude to form ortho mormal vectors. is an orthonormal basis such that [7] B is diagonal matrix B 11 {(8) (8) (8) CS Scanned and CamScanner