Vector space/K^n component wise/Example

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

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

with componentwise addition and with scalar multiplication given by

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

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