We know that given a set of scalars $\{c_1,c_2,\dots,c_n\}$ in $F$, there is a set of Lagrange polynomials associated with it. And since these are a basis of $\mathsf{P}_n(F)$, so it's clear that given a $g$ in $\mathsf{P}_n(F)$, there is a unique collection of parameters, say $b_1, b_2,\dots,b_n$ which are scalars, such that:
$$g=\sum^{n}_{i=0}b_if_i\ .$$
Now I guess it can be proved that: Given $g$ and a collection of parameters $b_1,b_2,\dots,b_n$, there is a unique set of Lagrange polynomials associated with it.
My current work: I'm not sure whether turning a set of parameters of $g$ into those $b_i$ for a given set of Lagrange polynomials is a linear trasformation. But if it is, it seems like it can be proved using a theorem in the book of linear algebra I'm reading...
Hint: This cannot be true because $b_i=g(c_i)$ and $g$ does not have to be injective. Try $g(x)=x^2$ and $b_1=1$ and $b_2=2$.