I've been given a vector space of linear polynomials in x, $p(x)=ax+b,\;\;$ $q(x)=cx+d$, and the inner product is defined to be $\langle p,q\rangle=ac+bd$. I've been able to verify all the axioms for the inner product except for the complex conjugate one, $\langle p,q\rangle^*=\langle q,p\rangle$, where $p$ and $q$ are vectors.
The issue I'm having is that I don't understand how the complex conjugate can apply if there isn't an $i$ in the equation. All I know is that the complex conjugate takes the form $(ax+iy)^*=ax-iy$, but I'm really confused as to what this means without $i$.
What is the complex conjugate of a vector that doesn't have an imaginary component?