Hello I wanted to know if someone can correct this exercise?

Let n be an integer greater than or equal to 2, and let a1, . . . , an be real numbers, all distinct. For any integer 1 k n, we consider the polynomial:

Lk(X) = (X - ai) / (ak - ai) where the product ranges over all i k from 1 to n.

For 1 k n, show that Lk is the unique polynomial P in Rn1[X] such that for all 1 k n: P(ai) = { 0 if i k, 1 if i = k }

We consider the mapping: F: Rn1[X] Rn P (P(a1), ..., P(an))

a. Prove that F is a linear mapping. b. Let (e1, ..., en) be the canonical basis of Rn. For 1 k n, show that there exists a polynomial P in Rn1[X] such that F(P) = ek. c. Prove that F is surjective, and then justify that F is bijective.

Let f be a function from R to R. a. Show that there exists a unique polynomial P Rn1[X] such that for all 1 k n: P(ak) = f(ak). This polynomial P is called the interpolation polynomial of f at the abscissas a1, . . . , an. b. Express the interpolation polynomial of f at the abscissas a1, ..., an using the polynomials L1, ..., Ln, and the values of f at a1, ..., an.