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). Be careful, because the last time I submitted this question, the answers were wrong.

Answer 1

Given that

Diagonal matrix : It is a special kind of symmetric matrix. It is a symmetric matrix with zeros in the off-diagonal elements. Two diagonal matrices are shown below.

A =

1 0 0 3

B =

5 0 0 0 3 0 0 0 1

Note : the diagonal of a matrix refers to the elements that run from the upper left corner to the lower right corner.

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