Definition:Dimension (Linear Algebra)

Module
Let $G$ be a unitary $R$-module which has a basis of $n$ elements.

Then $G$ is said to have a dimension of $n$ or to be $n$-dimensional.

A module is finite-dimensional if it is $n$-dimensional for some $n \in \N^*$.

The dimension of a unitary $R$-module $G$ is denoted $\dim \left({G}\right)$.

Vector Space
Given a vector space $V$, the dimension of $V$ is the number of vectors in a basis for $V$.

From Bases of Finitely Generated Vector Space, all bases of $V$ have the same number of elements.

The dimension of a finite-dimensional $K$-vector space $V$ is denoted $\dim_K \left({V}\right)$, or just $\dim \left({V}\right)$.

Alternatively, the dimension of $V$ can be defined as the maximum cardinality of a linearly independent subset of $V$.

The equivalence of these definitions follows from Linearly Independent Subset of Finitely Generated Vector Space.