Question

Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(v) (To show two functions are equal, show that they agree on all inputs) (b) Let V = C([a, b) and let el denote the unique positive constant function in V such that e Find an explicit formula for e in terms of a and b. If (,) is the L2 inner product on C(la, b]) (that is, 〈f,g)-J:f(x)g(z)dr), show that projei f is the unique constant function with the same average value as f on [a, b You may use facts from calculus (c) Let V = C([-a, a]) for some a > 0. Using the L2 inner product, V is an inner product space. Let E = {f ε V : f is even) and let O = {f V : f is odd }. (Recall the f is even if f(x) = f(-r) for all z in the domain, and odd if f(x)--f(-r) for all z in the domain.) Prove that E 0 = V, that E = OL and that O = E 1S

Answer 1

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