Assume that $V = P(3)$, Describe a basis for $V^*$ and express the linear functional $f : V \to \Bbb R$ given by $f\left(a_3x^3 + a_2x^2 + a_1x + a_0\right) = 2a_3 + a_2 – 5a_0$ as a linear combination of the basis you found for $V^*$.
I am struggling to find the basis for $V^*$, I know that the standard basis for $P(3)$ is $\left\{1, x, x^2, x^3\right\}$ and then the basis for $V^*$ is $f\left\{1, x, x^2, x^3\right\}$. Any help would be greatly appreciated, thank you.
Best Answer
You need to find a basis for the dual space $V^*$, $\{f_1,f_2,f_3,f_4\}$ such that $$f_i(v_j)=\begin{cases}1 \ \ \ \mbox{if} \ \ i=j \\0 \ \ \ \mbox{if} \ \ i\ne j \end{cases}. $$ As a small hint, try using a combination of derivatives and evaluation maps.