Let V = P2 and define a pairing of poiynomials p and r; in [,2 by (19.9) == 1I(0)q(0) +p(1)r1(1) +p(2)q(2). (a) (7 points) Show that this formula...

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5. Let V = P2 and define a pairing of poiynomials p and r; in [,2 by (19.9) == 1I(0)q(0) +p(1)r1(1) +p(2)q(2). (a) (7 points) Show that this formula defines an inner product on Pg. (1)) (4 points) Show that the polynomials p(t) = 4t2 — 8t + 2 and q(t) = —t2 — t + 4 areorthogonal with respect to this inner product. (0) (5 points) Find the matrix C that represents this inner product with respect to theordered basis S = (t2,t,1). (d) (4 points) Use coordinate vectors and your answer in part (c) to againshow that 4t2 — 8t + 2 and —t2 w t + 4 are orthogonal.