Definition:Vector Space on Cartesian Product

Theorem
Let $\struct {K, +_K, \times_K}$ be a division ring.

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

Let $+: K^n \times K^n \to K^n$ be defined as:
 * $\tuple {\alpha_1, \ldots, \alpha_n} + \tuple {\beta_1, \ldots, \beta_n} = \tuple {\alpha_1 +_K \beta_1, \ldots, \alpha_n +_K \beta_n}$

Let $\times: K \times K^n \to K^n$ be defined as:
 * $\lambda \times \tuple {\alpha_1, \ldots, \alpha_n} = \tuple {\lambda \times_K \alpha_1, \ldots, \lambda \times_K \alpha_n}$

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

Also see

 * Vector Space on Cartesian Product is Vector Space