Let $A=\{0,1,2,3\} \subset \mathbb{R}$ and $\mathbb{R}[X]$ be the set of all polynomials in one variable over $\mathbb{R}$. Let $V$ be the $\mathbb{R}$-vector space of functions $f: A \rightarrow \mathbb{R}$ such that there exists a polynomial $P \in \mathbb{R}[X]$ with $f(a)=P(a)$ for all $a \in A$. Determine the dimension of the $\mathrm{R}$-vector space $V$. (a) 2 (b) 3. (c) 4. (d) $\infty$.
I tried solving this problem by seeing that there always exists a Lagrange polynomial of degree 3 that will do our job for the part that $f(a)=P(a)$. And since the $\mathbb{R}$ vector space V consists of all the arbitrary functions, the dimension should be infinite.