I am taking an linear algebra class and we are on inner product space. I have a question that wants us to evaluate $T^*$ at a given vector. This is not a homework question, it is a textbook question, specifically question 3c in 6.3 of Linear Algebra by Friedberg, Insel, Spence (pg 366). I will it write out though:
$V=P_1(R)$ (the set of polynomials with degree $\leq 1$ with real coefficients) with $\langle f,g\rangle=\int_{-1}^1f(t)g(t)dt,$ $T(f)=f'+3f.$ Compute $T^*(f(t))$ where $f(t)=4-2t$.
I know the answer (it's in the textbook) but I am not sure how to obtain the solution. It's my understanding that since $V$ is finite-dimensional we know there is an adjoint and that for all $f,g\in P_1(R)$ $\langle T(g),f\rangle=\langle g,T^*(f)\rangle$. I tried setting $f=4-2t$ and I thought maybe set $g=1$ to make my computation easier but I don't think I am getting this right. I also decided to take the standard basis for $P_1(R)$ and used Gram-Schmidt to obtain an orthonormal basis $\gamma = \{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}}x\}$. I am not really sure what to do next, like do I need to calculate the exact adjoint map or something. Any help would be greatly appreciated.
UPDATE: I figured out based on the comment made by darij grinberg. I used the fact that $\langle T(g),f\rangle=\langle g,T^*(f)\rangle$ where $g=1$ and $g=t$ and setting $T^*(f) = a+bt$ and figuring out $a$ and $b$ and then using that to compute $T^*(4-2t)$.