Let P 3 have the inner product given by evaluation at 3, 1, 1, and 3 Let p 0 (t) 1, p 1 (t) t, and p 2 (t) t 2 a Compute the orthogonal projection of P 2 onto the sub space spanned by p 0 and p 1 b Find a polynomial q that is orthogonal to p 0 and p 1 , such that p 0 p 1 q is an orthogonal basis for Span p 0 , p 1 , p 2 Scale the polynomial q so that its vector of values at ( 3, 1, 1, 3) is (1, 1, 1, 1)

Question: Let P 3 have the inner product given by evaluation at -3,-1, 1, and 3. Let p 0 (t) = 1, p 1 (t) =

Let P₃have the inner product given by evaluation at -3,-1, 1, and 3. Let p₀(t) = 1, p₁(t) = t, and p₂(t) = t².

a. Compute the orthogonal projection of P²onto the sub - space spanned by p₀and p₁.

b. Find a polynomial q that is orthogonal to p₀and 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).

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