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.
The matrix representation of T is below;© BrainMass Inc. brainmass.com March 4, 2021, 8:12 pm ad1c9bdddf
The explanations are in the attached pdf file.
UPDATE FROM OTA, CORRECTED RESPONSE:
Substitute x = sum a_i t^i and y = sum b_k t^k into ...
Linear Transformations, Hilbert Space, Inner Product and Matrix Adjoint are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.