Question

A] Consider the inner product space obtained by equipping ?[0,2] with the inner product given below:
〈?(?),?(?)〉 = ∫ ?(?)?(?)?? 2 0


Determine the value of each of the following (simplifying where possible; no decimals). You do not have to show the steps of calculating the integrals, but must at least write the integrals used in your calculations.   
(A.3) ?(??,?? + 10), i.e. the distance between ?? and ?? + 10 .   







(A.4) Determine the value of ? so that the functions ?(?) = ? and ?(?) = ?? + 2 are  orthogonal relative to the given inner product.

(A.5) Consider the subspace ℙ1 of ?[0,2] consisting of all polynomials of degree 1 or less. Use the Gram-Schmidt process to convert its standard basis ? = {1,?} to an orthonormal basis.

(A.6) Obtain the polynomial ?(?) ∈ ℙ1 that “best approximates” ?(?) = ?2 in the sense that ?(?(?),?(?)) is minimized. Express your final answer in the form ?(?) = ?? + ? where ? and ? are simplified as much as possible (no decimal approximations).

Answer 1

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