Let ℙ3 have the inner product given by evaluation at −3, −1,
1, and 3. Let p0(t)=1, p1(t)=2t, and p2(t)=t4.
a. Compute the orthogonal projection of p2
onto the subspace spanned by p0
and p1.
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).
Extracted text: Let P, have the inner product given by evaluation at - 3, - 1, 1, and 3. Let po (t) = 1, p, (t) = 2t, and p2(t) = t*. a. Compute the orthogonal projection of p, onto the subspace spanned by po and P1. b. Find a polynomial q that is orthogonal to po and p,, such that {P0•P1,q} is an orthogonal basis for Span {Po.p1.P2. Scale the polynomial q so that its vector of values at (- 3, – 1,1,3) is (1, – 1, – 1,1).