# Inner product over complex vector space and conjugation

It is commonly stated (for example here) that

$\langle v |w \rangle=\overline{\langle w |v \rangle}$

where $v$ and $w$ are vectors over complex vector space, $\langle.|.\rangle$ is the inner product and $\overline{z}$ is the complex conjugate. But I find that this relation simply doesn't hold, e.g. if $v := (1+2i,3+4i)$ and $w := (5+6i,7+8i)$, then

$\langle v |w \rangle=-18+68i$ and $\overline{\langle w |v \rangle}=-18-68i$

so rather than the original relation, $\langle v |w \rangle=\langle w |v \rangle$ seems to hold. Obviously I am missing something, maybe I'm at fault assuming that the inner product in this case is identical to the dot product, but I'm not sure.

For $v=(v_1,v_2)$ and $w=(w_1,w_2)$ in $\mathbb C^2$ the standard inner product is $$\langle v,w \rangle = v_1 \overline{w_1} + v_2 \overline{w_2}$$ From this definition, the identity $$\langle v,w \rangle=\overline{\langle w,v \rangle}$$ obviously holds.