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 $\left[{1 \,.\,.\, n}\right] \subset \N^*$.

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

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 26$: Example $26.1$