Vector Space on Cartesian Product is Vector Space

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

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

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

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

Proof 1

This is a special case of the Vector Space of All Mappings, where $S$ is the set $\closedint 1 n \subset \N^*$.


Proof 2

This is a special case of a direct product of vector spaces where each of the $G_k$ is the $K$-vector space $K$.