Let $n$ be a \reftext{def:natural-numbers-2026a}{natural number} and let $\mathbb{C}^{n}$ be the \reftext{def:complex-coordinate-space-cn-2026a}{complex coordinate space}, with the componentwise operations of that definition.

Then $\mathbb{C}^{n}$ with these operations is a \reftext{def:vector-space-2026a}{complex vector space}. Its \reftext{lem:vector-space-basic-identities-2026a}{zero vector} is the $n$-tuple $0_{\mathbb{C}^{n}}=(0,\dots,0)$ all of whose components are the complex number $0$, and the additive inverse of $u=(u_{1},\dots,u_{n})$ is $-u=(-u_{1},\dots,-u_{n})$.