Chapter 6.7, Problem 10E

Let ℙ₃ have the inner product as in Exercise 9, with p₀, p₁, and q the polynomials described there. Find the best approximation to p(t) = t³ by polynomials in Span {p₀, p₁, q}.

9. Let ℙ₃ have the inner product given by evaluation at −3, −1, 1, and 3. Let p₀(t) = 1, p₁(t) = t, and p₂(t) = t².

1. a. Compute the orthogonal projection of p₂ onto the subspace spanned by p₀ and p₁.
2. b.Find a polynomial q that is orthogonal to p₀ and p₁, such that {p₀, p₁, q} is an orthogonal basis for Span {p₀, p₁, p₂}. Scale the polynomial q so that its vector of values at (−3, −1, 1, 3) is (1, −1, −1, 1).