Definition 1

The dimension of a vector space $E$ over a field $K$, denoted by $\dim_K E$ or $[E:K]$ (or simply $\dim E$) is the cardinal of any of the bases of $E$. If $M$ is a subset of $E$, the rank of $M$ (over $K$*), denoted by* $\operatorname{rg} M$ or $\operatorname{rg}_K M$, is the dimension of the vector subspace of $E$ generated by $M$.

To say that $E$ is finite-dimensional is equivalent to saying that $E$ is a $K$-module of finite length and $\dim_K E = \operatorname{long}_K E$.