Vector Space

A vector space over a field $K$ is a non-empty set $V$ together with a binary operation $+$ and a binary function $\cdot$ from $K \times V$ to $V$ such that:

• $(V, +)$ is an abelian group
• for all $\alpha \in K$, the map $\alpha \mapsto (x \mapsto \alpha \cdot x)$ is a ring homomorphism from $K$ to $\text{End}V$.