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In this question we denote by P2(R) the set of functions {ax2 + bx + c...

In this question we denote by P2(R) the set of functions {ax2 + bx + c : a, b, c ∈ R}, which is a vector space under the usual addition and scalar multiplication of functions. Let p1, p2, p3 ∈ P2(R) be given by p1(x) = 1, p2(x) = x + 2x 2 , and p3(x) = αx + 4x 2 . a) Find the condition on α ∈ R that ensures that {p1, p2, p3} is a basis for P2(R). (You are free to assume that the functions 1, x and x 2 are linearly independent.) b) In the case that α = 5, write the function p(x) = 1 + x + x 2 as a linear combination of p1, p2 and p

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