- #1
wtmoore
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Homework Statement
I have this question that I am trying to figure out about orthonormality,I have tried to take a picture of it and put it on here but I can't figure out the url. Anyway I will try and write it out.
Show that the vector {sin(x),cos(x)} is a basis for the vector space defined by:
V={asin(x) + bcos(x) l a,b ε ℝ, 0≤ x ≤ pi} using the inner product :
<f,g>=∫(0 to pi)fgdx, fgεV
and determine an orthonormal basis.
Homework Equations
The Attempt at a Solution
I found the integral of sin(x)cos(x)dx between 0 and pi to be 0.
This make it orthonormal right as it's the same as the dot product.
Now I think I have to find out whether <f,f> is = 1 (of unit length)
so I did integral of sin^2(x)dx between 0 and pi but found pi/2. I also found the same for cos^2(x).
Does this mean they are not orthonormal? I don't know if it makes a difference that <f,f>=<g,g>.
Thanks