Vector Space on Cartesian Product is Vector Space

Theorem

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 $\closedint 1 n \subset \N^*$.

$\blacksquare$

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 $K$.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.1$
• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 32$. Definition of a Vector Space: Example $62$