The Complex Coordinate Space is a Complex Vector Space

Let $n$ be a natural number and let $\mathbb{C}^{n}$ be the complex coordinate space, with the componentwise operations of that definition.

Then $\mathbb{C}^{n}$ with these operations is a complex vector space. Its 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 $-u=(-u_{1},\dots,-u_{n})$.