Let $V=\mathbb R_2[x]$ be a vector space. Let us define the inner product of $V$ as: $$\langle p,q\rangle=\int_0^1p(x)q(x)\,dx$$
Let $W=\mathbb R$ be a space and $T:V\to W$ a linear transformation for which $T(p(x)) = p(0)$. How do I find $T^*$?
I've seen that this question was asked a few times on this site but couldn't find a question that is similar in mine (as in a similar linear transformation and vector space).
I've defined $T$ as $T=(1,0,0)$ over the standard base $\{1,x,x^2\}$, and I know I need to look for $T^*$ for which this equation holds: $\langle Tp,q\rangle =\langle p,T^*q\rangle \forall p,q \in \mathbb R_2[x]$.
Would love to see a solution to this one, thanks!