Linear Algebra Problems

Let Pn denote the vector space of polynomials of degree < n in the variable x. Let B and C be the bases for P3 and P2 given by B = {x³ - 3x, 3x2 + x, x2 - x, 1 + 2x} C = {x2 - 1,1,2x} Find the matrix of the linear transformation d P3 P2, f(x) f' (x) with respect to the bases B and C Theorem 1. Any nonzero polynomial f(x) = ao + aix + anx of degree at most n has at most n roots. In other words, if there are n + 1 distinct numbers bo, b1, bn and f(bo) = f(b1) = = f (bn) = 0 then f (x) is the zero polynomial. Assume xo, T1, In are given distinct real numbers. Define the operator T: Pn Rn+1 by p(xo) p(x1) T(p(x)) = p(xn) It can be shown (you don't have to do this) that T is a linear operator (for example: problem 10 in section 5.4 does this with xo = - -3,x1 = - 1, x2 = 1, x3 = 3 and n = 3.) (a) Show the operator T is one-to-one (injective). (b) Conclude that T is also onto (surjective). (c) Let us define the matrix 1 xo x2 n xo 1 X1 x2 1 xn n 1 A = 1 x2 x2 x2 n : : 1 In xn x n n Explain how A relates to T and show that A is invertible. Hint: showing A is invertible is quite difficult without explaining how A and T are related: I don't advise you to try it!