Proving polynom existence

Question

Aloha,

Let $x_i, y_i \in K $ where K is a field and let $x_1,...x_2$ be disjoint. Prove that there existis exactly one polynom $f\in K[t]$ with deg(f)$\leq$n-1 und $f(x_i) = y_i$ with $i=1,...,n$.

So for the proof: Let be f,q be two polynomials with the given attribute, so deg(f-q)$\leq$ n but with the n+1 root thus f-q is the nullpolynom. Now I need to prove existence.

Thanks in advance.

Answer 1

"Approach": $f(x)=a_0+a_1x+...+a_nx^n$. From $f(x_i)=y_i$ you get a linear sytem

(*) $a_0+a_1x_i+...+a_nx_i^n=y_i$ ($i=1,...,n$)

with a Vandermonde - matrix ( -> Google). This matrix is invertible.

This shows existence and uniqueness.