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A subspace spanned by polynomials 1 and x 
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#1
Dec2109, 08:29 PM

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1. Let W be the linear subspace spanned by the polynomials 1 and x. Find an orthogonal projection of the polynomial p(x) = 1+x^2 to W. Find a basis in the space W(perp)
My problem is that I don't know how to represent W as a matrix so that I could apply the orthogonal projection formula. Would it simply be [1, x] or [1 ; x]? Thanks! 


#2
Dec2109, 09:46 PM

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P: 25,228

{1,x} is certainly a basis for W. But to define orthogonal you need to say what the inner product is.



#3
Dec2109, 10:26 PM

P: 1,089

Remember, the dot product (or inner product) should be equal to 0 if they are orthogonal.



#4
Dec2109, 10:53 PM

P: 5

A subspace spanned by polynomials 1 and x
(This is not my homework by the way, I'm studying for my final exam which is tomorrow, but I just can't figure out this problem.) Thank you. 


#5
Dec2109, 11:12 PM

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If a*1+b*x is a point in W that is the orthogonal projection, you want that <(1+x^2)(a*1+b*x),1>=0 and <(1+x^2)(a*1+b*x),x>=0. That would say that the difference between (1+x^2) and (a*1+b*x) is perpendicular to all of the vectors in W which is the span of {1,x}, wouldn't it? It's just two linear equations in the unknowns a and b.



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