---- > [!definition] Definition. ([[dimension]]) > The **dimension** of a [[vector space]] $V$ over [[field]] $k$ is its [[rank of a free module|rank]] as a [[free module|free]] $k$-[[module]], i.e., the [[cardinality]] of a (and therefore any) [[basis|basis]] of $V$. It is denoted $\text{dim}_{k} \ V$. ^definition > [!specialization] Statement for finite-dimensional vector spaces. > The **dimension** of a [[vector space#Finite-Dimensional Vector Space|finite-dimensional vector space]] $V$ is the length of any [[basis]] of $V$. We denote it $\dim V$. > > That this notion is [[well-defined]] admits an easier proof than the general case: [[finite-dimensional vector space basis length does not depend on basis]]. See also [[dimension of a subspace]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```