The fact that two polynomial are equal $\iff$ their coefficient are equal is a definition or a proposition? Let $$p(X)=a_0+a_1X+...+a_nX^n\quad \text{and}\quad q(X)=b_0+b_1X+...+b_nX^n.$$
In the vector space of polynomial with coeeficient in $\mathbb R$, the fact that $$p(X)=q(X)\iff \forall i=0,...,n, a_i=b_i,$$
is a a proposition or a definition ? The reason is that I tried to proof that $\{1,X,...,X^n\}$ is a basis of polynomial of degree less or equal than $n$. So, I'm not so sure if there is I can prove that $$a_0+a_1X+...+a_nX^n=0\implies a_i=0\text{ for all $i$},$$
or if this is simply the definition or if I have/can prove that rigorously.
 A: When answering this question, we run into a terminological problem: the word polynomial is used for two similar but distinct things. If $R$ is (say) a ring, we define the ring $R[X]$ of polynomials over $R$: the objects are formal expressions of the form $\sum_{i=0}^n a_i X^i$. In this case, the fact that two polynomials are equal if and only if their coefficients are equal is true by definition.
Now if $F(X) \in R[X]$ is a polynomial, and $r \in R$, then we obtain an element $F(X/r) \in R$ by substituting occurrences of $X^i$ with $r^i$ and using the operations of $R$ to evaluate the resulting expression. Thus, given a polynomial $F(X) \in R[X]$, we get a function $\hat F: R \to R$ with $\hat F(r) = F(X/r)$. We will also refer to the set of all functions of this form as polynomials.
However, depending on the ring $R$, two different polynomials $F, G$ may give rise to the same function $\hat F, \hat G$. For example, if $R = \mathbb F_2$, then $F(X) = 0$ and $G(X) = X^2 - X$ give the same function. So if you use the word "polynomial" for the function (rather than the formal expression), in this case it is not true that polynomials are equal if and only if they have the same coefficients.
For special rings, however, it may still be true: in that case you may be able to prove that if $F \neq G$ then $\hat F \neq \hat G$. For example, this is true whenever $R$ is an infinite field. In that case, the fact that two polynomials are equal if and only if their coefficients are equal becomes a theorem rather than a definition.
A: It's a definition. Note that you have to differ between polynomials and polynomial functions. Over finite fields, the latter may be represented by multiple different polynomials, e. g $p, q$. Here, $p$ and $q$ are not equal as polynomials, but are equal as polynomial functions.
