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Linear Transformations : Hilbert Space and Inner Product

Please see the attached file for the fully formatted problems.

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. Define a linear transformation S as

Show that the subspace P5 is invariant under S. That is, verify that S maps members of P5 back into P5. Define T as the restriction of S to P5. (This part is already done. So I just send it for information.)

The matrix representation of T is below;

My question is:

And, if we define y(t) as

y(t)=1 + t + t2 + t3 + t4 what is the optimal value of that solves the approximation problem?

(Hint: T in min operator is not the matrix defined above. But, it is the operator defined above as Sx. So wherever we see S we can use T. (Sx=Tx) And U is just the space of polynomials.)

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As the course on polynomials requires the ability to calculate an inverse of a matrix as a prerequisite, and you have not ...

Solution Summary

Linear Transformations, Hilbert Space and Inner Products are investigated.

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