# Definition:Vector Space on Cartesian Product

## Theorem

Let $\struct {K, +, \circ}$ be a division ring.

Let $n \in \N_{>0}$.

Let $+: K^n \times K^n \to K^n$ be defined as:

$\tuple {\alpha_1, \ldots, \alpha_n} + \tuple {\beta_1, \ldots, \beta_n} = \tuple {\alpha_1 +_R \beta_1, \ldots, \alpha_n +_R \beta_n}$

Let $\times: K \times K^n \to K^n$ be defined as:

$\lambda \times \tuple {\alpha_1, \ldots, \alpha_n} = \tuple {\lambda \times_R \alpha_1, \ldots, \lambda \times_R \alpha_n}$

Then $\struct {K^n, +, \times}_K$ is the $K$-vector space $K^n$.