Let $\mathbf{F}$ be a field with at least 4 elements and $\mathbf{V}$ be the vector space over $\mathbf{F}$of all polynomials of degree $\leq $3
For $a\in \mathbf{F}$, define $f_a\colon \mathbf{V}\to \mathbf{F}$ by $f_a(p):=p(a)$.
If $a_1$, $a_2$, $a_3$, $a_4$ are distinct members of $\mathbf{F}$, how would you show that the set $\{f_{a_1}, f_{a_2}, f_{a_3}, f_{a_4}\}$ is a basis for $\mathbf{V}^*$ (the dual space of $\mathbf{V}$)? I have found that the set is linearly independent, however am struggling on proving it spans V*
Any help greatly appreciated.