Let $ f \in \mathbb R_{\leq4}[t]$ such that $f(p)=f \bigg( \sum\limits_{k=0}^4 a_k t^k\bigg):=f \bigg( \sum\limits_{k=1}^4 k a_k t^{k-1}\bigg)$ and define $\langle p,q \rangle := \int\limits_{-1}^1 p(t)q(t)dt$. I am asked to find $\operatorname{Im} f^{ad}$ and $\operatorname{Null} f^{ad}$.
I started by calculating an orthonormal basis $B: b_0=\frac{1}{\sqrt 2}, b_1=\sqrt\frac{3}{2}t$, and so on. My goal was to find $[f]_{B,B}=[\operatorname{id}]_{A,B}[f]_{A,A}[\operatorname{id}]_{B,A}$, where $A=\{1,t,t^2,t^3,t^4 \}$ and to transpose it in order to find $[f^{ad}]_{B,B}$ and then transform it to $[f^{ad}]_{A,A}$ where $\operatorname{Im} f^{ad}$ and $\operatorname{Null} f^{ad}$ could be easily seen.
Yet it turns out to be a mess with all the fractions, square roots and matrix inversions. That made me doubt my approach and wonder if there is a more elegant one.