Let n be a nonnegative integer, and let a1...an+1 be distinct real numbers. Let Pn be the vector space of real

Question

polynomials f(x) of degree <= n. Define F: Pn --> R^n+1 by f -> (f(a1)....f(an+1))
b) Explicitly find F−1(e1), . . . , F−1(en+1) (where e1, . . . , en+1 are the standard basis
vectors in Rn+1) in the case n = 2 and aj = j for j = 1, 2, 3. [Hint: Where does
(x − a)(x − b) vanish?]
c) in general, deduce that f-1(e1)...f-1(en+1) form a basis of Pn. In the special case done in (b), express the polynomial x as a linear combination of these basis elements

Answer 1

Your notation is confusing. I assume by f-1 you mean f inverse, but you could do better.

So you're mapping polynomials to their values on a fixed set of points. The hint and suggested proof tell you that the inverse of a vector all of whose values except 1 are 0 is a polynomial that has n known (and distinct roots), and whose nonzero value for some other x is also known. That is enough information to specify the polynomial precisely.

There. I've expanded the hint for you to get you started.