Let $V=P_2(\mathbb{R})$ with basis $B= \{1-2X,3+X,8X^2\}$. Let $T:V\rightarrow V$ be given by $T(aX^2+bX+c)=bX^2-aX+1$. Assume $T$ is linear. Find the dual basis $B^*=\{f_1,f_2,f_3\}$ by finding for each $i$ a formula for $f_i(aX^2+bX+c)$, and find $T^t(f_3)$
I'm having trouble doing this problem. So far I know that I have to set up each $f_i=0$ and solve for it, but I am getting elements that are in $V$ and not $V^*$. Thanks in advance for any help.