Let V be the space of real polynomials of degree at most 1 with inner product defined by $\langle p,q \rangle = \frac12\int_{-1}^1p(t)q(t)dt.$ Define $\alpha\in End(V)$ by $\alpha(p)=p(0)+p(1)t.$ Find the adjoint endomorphism $\alpha^*.$
For this problem, I am wondering about several things. First, I know that $\langle\alpha(p),q\rangle= \langle p,\alpha^*(q)\rangle$. I wasn't sure if I could assume that $\alpha^*(q)=r+st$ for constants $r,s$. But I think I can since the adjoint is also an endomorphism. So, I attempted to compute $\alpha^*(q)$ by first defining $q(t)=m+nt,$ and I found $\alpha^*(q)=(m+\frac13n)+nt$. If this is even correct, I wouldn't know how to turn this into an expression for $\alpha^*(q)$ in the way that $\alpha(p)$ is defined.
Or should I determine a matrix representation for $\alpha$ and take its conjugate transpose? If so, how does that new matrix translate to an expression for $\alpha^*(q)$?
Thank you for your time in helping.