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Linear Transformations, Hilbert Space, Inner Product and Matrix Adjoint

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Please see the attached file for the fully formatted problems.
Find a 5x5 matrix M>0 such that if and x(t)= then

Can we use this definition to find the adjoint of T (T is given at the end)?

This part is the additional information to solve the question above;

Let V=L2[-1,1] be the Hilbert space of functions over the time interval [-1,1] with inner product

Let P5 V be the subspace of polynomials of order 4 or less, endowed with the inner product and norm of V, and let , be its natural basis. The linear transformation S is defined as

T is defined as the restriction of S to P5.

Sx=Tx

The matrix representation of T is below;

https://brainmass.com/math/linear-transformation/152120

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UPDATE FROM OTA, CORRECTED RESPONSE:
(a1)
Substitute x = sum a_i t^i and y = sum b_k t^k into ...

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