From Axler's Linear Algebra Done Right, chapter 4, exercise 5:
Suppose is a nonnegative integer, are distinct elements of
, and .
Prove that there exists a unique polynomial
such that for
[This result can be proved without using linear algebra. However, try to
find the clearer, shorter proof that uses some linear algebra.]
If then as is at most degree ,
there are at most distinct 's such that .
Equivalently, if had at least distinct
's such that , then . Letting
then the previous statement implies that ,
so is injective. As is square, [Axler 3.69] implies that
is invertible. Hence, if
then the polynomial is the unique
polynomial such that for .