Let x1, x2, and x3 be distinct real numbers such that x1 < x2 < x3 and let T : P2 R3 be
Chapter 8, Problem 21(choose chapter or problem)
Let x1, x2, and x3 be distinct real numbers such that x1 < x2 < x3 and let T : P2 R3 be the function defined by the formula T(p(x)) = p(x1) p(x2) p(x3) (a) Show that T is a linear transformation. (b) Show that T is one-to-one. (c) Verify that if a1, a2, and a3 are any real numbers, then T 1 a1 a2 a3 = a1P1(x) + a2P2(x) + a3P3(x) where P1(x) = (x x2)(x x3) (x1 x2)(x1 x3) P2(x) = (x x1)(x x3) (x2 x1)(x2 x3) P3(x) = (x x1)(x x2) (x3 x1)(x3 x2) (d) What relationship exists between the graph of the function a1P1(x) + a2P2(x) + a3P3(x) and the points (x1, a1), (x2, a2), and (x3, a3)?
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