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A problem on finding orthogonal basis and projection

by visharad
Tags: basis, orthogonal, projection
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Nov21-11, 10:23 AM
P: 53
Use the inner product <f,g> = integral f(x) g(x) dx from 0 to 1 for continuous functions on the inerval [0, 1]

a) Find an orthogonal basis for span = {x, x^2, x^3}

b) Project the function y = 3(x+x^2) onto this basis.
I know the following:
Two vectors are orthogonal if their inner product = 0
A set of vectors is orthogonal if <v1,v2> = 0 where v1 and v2 are members of the set and v1 is not equal to v2
If S = {v1, v2, ..., vn} is a basis for inner product space and S is also an orthogonal set, then S is an orthogonal basis.

Regarding projection, I know that if W is a finite dimensional subspace of an inner product space V and W has an orthogonal basis S = {v1, v2, ..., vn} and that u is any vector in V then,
projection of u onto W = <u, v1> v1/||v1||^2 + <u, v2> v2/||v2||^2 + <u, v3> v3/||v3||^2 + ...<u, vn> vn/||vn||^2

I can calculate integrals, but I really do not know how to fit all these together for this problem. I am not sure how to start.
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