Vector Space

A vector space over a field $F$ is a set $V$ with an addition making $(V, +, \mathbf{0})$ an abelian group, together with a scalar multiplication $F \times V \to V$ satisfying compatibility, distributivity, and $1 \cdot \mathbf{v} = \mathbf{v}$.

A vector space is determined up to isomorphism by its dimension — the cardinality of any basis. Two finite-dimensional vector spaces over the same field are isomorphic if and only if they have the same dimension.

When the scalars come from a ring rather than a field, the structure is called a module. Modules are more general and less well-behaved than vector spaces.