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(Solved): Problem 6 (Orthogonal Complement). Consider the vector space P_(2) along with the inner product (:f( ...
Problem 6 (Orthogonal Complement). Consider the vector space P_(2) along with the inner product
(:f(x),g(x):)=\int_(-1)^1 f(x)g(x)dx
(a) Find the orthogonal complement of W_(1)=span(1+x).
(b) Find the orthogonal complement of W_(2)=span(1+x,x^(2)).
(c) Verify that .
(d) The observation made in (c) is true in full generality: prove that if W_(1)subeW_(2) are two subspaces of an
inner product space, then .