Vector space/K^n component wise/Example

Let ${\displaystyle {}K}$ denote a field, and let ${\displaystyle {}n\in \mathbb {N} _{+}}$. Then the product set

${\displaystyle {}K^{n}=\underbrace {K\times \cdots \times K} _{n{\text{-times}}}={\left\{(x_{1},\ldots ,x_{n})\mid x_{i}\in K\right\}}\,,}$

with componentwise addition and with scalar multiplication given by

${\displaystyle {}s(x_{1},\ldots ,x_{n})=(sx_{1},\ldots ,sx_{n})\,,}$

is a vector space. This space is called the ${\displaystyle {}n}$-dimensional standard space. In particular, ${\displaystyle {}K^{1}=K}$ is a vector space.