Definition:Vector Space on Cartesian Product

Theorem
Let $$\left({K, +, \circ}\right)$$ be a division ring.

Let the $K$-module $$K^n$$ be defined as in Module on Cartesian Product.

Then $$\left({K^n, +: \times}\right)_K$$ is a $K$-vector space.

This will be referred to as the $$K$$-vector space $$K^n$$.

Proof
This is a special case of Vector Space of All Mappings, where $$S$$ is the set $$\left[{1 \,. \, . \, n}\right] \subset \mathbb{N}^*$$.

It is also a special case of a Product of Vector Spaces where each of the $$G_k$$ is the $K$-vector space $$K$$.