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!
{1,x} is certainly a basis for W. But to define orthogonal you need to say what the inner product is.
If the inner product is given by: <p(x),q(x)> = integral (p(x)*q(x), x, 0, 1), how would i go about solving this problem? (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.
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.