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2. Find the dual basis of the basis 1,x,x2,x3, x of P4 (x). Hint: Find linear functionals 90,91,92,93,94 satisfying pi(a.+ax+ax + azx + a.x4 = a, for each i =0,1,2,3,4. If you use the hint, pleas justify. Of course, include all key details in your solution. 3. Suppose that 1,1.2.13 are three distinct real numbers. Define the linear functionals n.n.,23 on P2 (x) by ni (p) = p(1,), -1,2,3 . (a) Prove that is a basis for 2(x)'. (b) Find a basis of P2 (x) for which is its dual basis. (This is called the interpolation problem.)

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