Linear functionals and dual bases 
How do I tackle this question? I am a little hazy on linear functionals and integral signs.
 A: Denote $B := \{v_1, v_2, v_3\}$. You have to find a basis $B^*= \{ L_1, L_2, L_3\}$ for the dual space $V^{*}$ such that
$$L_i(v_j) = \begin{cases}1 & \mbox{if } i=j \\ 0 & \mbox{if } i \neq j \end{cases}.$$
Hint: Try combinations of evaluations, derivatives and integrals. 
The following two subjects might be helpful:
Finding a dual basis for the vector space of polynomials degree less than or equal to 2 
finding dual basis of vector space of polynomial degree less than or equal to 3
After that write $L$ in terms of $L_1$, $L_2$ and $L_3$.
A: An easier answer: define $f_0,f_1,f_2$ to be the dual basis to $1,x,x^2$.  So, $f_i$ are linear functionals with
$$
f_i(x^j) = \begin{cases}1 & i = j \\ 0 & i \neq j\end{cases}
$$
We note that $f_0,f_1,f_2$ is a basis of the dual space, so that
$$
L = a_0f_0 +  a_1 f_1 + a_2 f_2
$$
for some scalars $a_i$.  In fact, we may state that for $j = 0,1,2$,
$$
L(x^j) = a_0f_0(x^j) +  a_1 f_1(x^j) + a_2 f_2(x^j) = a_j
$$
So, we conclude that
$$
a_0 = L(1), \quad a_1 = L(x), \quad a_2 = L(x^2)
$$
