First of all let me assess that this is about a graded homework I have to do. Please prioritize developing on how I should begin to answer the question rather than a quick answer. Yes it's graded but I'd much rather understand it than just copy paste an answer.
So this is the second part of the exercice. In the first part, we proved that :
With $E=\mathbb{R}_3[X]$, $B = \{1+X,X+X^2,X^2+X^3,X^3\}$ is a base of E.
Now here is the question I'm having trouble with :
Let $l_1(f) = f(0),l_2(f) = f'(0),l_3(f) = f"(0),l_4(f) = f^{(3)}(0), \forall f \in E$ be 4 linear forms on $E$.
Show that $L=(l_1,l_2,l_3,l_4)$ is a base of $E^*$ the dual space of $E$.
I must confess I had some trouble 'visualizing' the concept of dual space but I think I got it down now. Still I'm not able to find how to prove $L$ is a base.
Thanks to everyone that might help, it's greatly apreciated :)