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 $$n \in \mathbb{N}^*$$.

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 $$\dim \left({V}\right)$$.

Manifold
Let $$M \ $$ be a manifold.

We say $$M \ $$ has dimension $$n \ $$ if, for every point $$x \in M, \exists U \in \vartheta_M$$ such that $$\vartheta_M \ $$ is the topology of $$M \ $$, $$x \in U$$, and there exists a homeomorphism $$\phi:U \to \mathbb{R}^n \ $$.

In the context of smooth manifolds and differential topology, the homeomorphism above is strengthened to a diffeomorphism.