The [[Complex Numbers|Complex]] [[Vector Space]] is an extension of the typically [[Real Numbers|Real Number]] [[Vector Space]] allowing [[Complex Numbers]].
$\huge \begin{align}
\mathbb{C}^{N} &= \left\{
\mat{
z_{0}\\z_{1}\\ \vdots
} : z_{i} \in \C
\right\}
\end{align}
$
The [[Field]] defined for the $\C^{N}$ is $\C$.
The [[Basis Vectors]] are the same as for $\R^N$, as our field is $\C$.
$\huge \begin{align}
\left\{
\mat{1 \\0\\ \vdots},
\mat{0 \\1\\ \vdots}
\right\}
\end{align}
$