Find scalar product on $\mathbb{R}_n[x]$ and polynomial q such that $\theta=(p,q)$

Question

I am working on solving an example, which is as follows: Linear transformation is defined by $\theta(p)=p'(0)$, find scalar product on $\mathbb{R}_n[x]$ and polynomial $q$ such that $\theta(p)=(p,q)$. This should be solved using the bilinear forms.

Do you know of any book or lecture pdfs on bilinear forms, that deals with similar problems, as I do not know where to even begin?

Answer 1

You can take the canonical scalar product of $\mathbb R_n\left[X\right]$ So you are looking for $q(X) = \sum_{k=0}^n q_k X^k$ and the scalar product matrix such that : \begin{align} \left\langle p, q\right\rangle &= p'(0) &\forall p\in \mathbb R_n\left[X\right]\\ \implies\sum_{i=0}^n\sum_{j=0}^n p_iq_j\left\langle X^i, X^j\right\rangle &= p_1\\ \implies\sum_{i=0}^n p_iq_i &= p_1.\\ \end{align}

So $q_1 = 1$ and $q_i = 0$ for $i\neq 1$. So $q(X) = X$.

This idea works fine for any other scalar product (See Riesz representation theorem).