Editing Dimension (vector space) (section)

{{short description|Number of vectors in any basis of the vector space}}
[[File:Dimension levels.svg|thumb|A diagram of dimensions 1, 2, 3, and 4]]
In [[mathematics]], the '''dimension''' of a [[vector space]] ''V'' is the [[cardinality]] (i.e., the number of vectors) of a [[Basis (linear algebra)|basis]] of ''V'' over its base [[Field (mathematics)|field]].<ref>{{cite book|last=Itzkov|first=Mikhail|title=Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics|publisher=Springer|year=2009|isbn=978-3-540-93906-1|page=4|url=https://books.google.com/books?id=8FVk_KRY7zwC&pg=PA4}}</ref><ref>{{Harvard citation text|Axler|2015}} p. 44, §2.36</ref> It is sometimes called '''Hamel dimension''' (after [[Georg Hamel]]) or '''algebraic dimension''' to distinguish it from other types of [[dimension]].

For every vector space there exists a basis,{{efn|if one assumes the [[axiom of choice]]}} and all bases of a vector space have equal cardinality;{{efn|see [[dimension theorem for vector spaces]]}} as a result, the dimension of a vector space is uniquely defined. We say <math>V</math> is '''{{visible anchor|finite-dimensional}}''' if the dimension of <math>V</math> is [[wiktionary:finite|finite]], and '''{{visible anchor|infinite-dimensional}}''' if its dimension is [[infinity|infinite]].

The dimension of the vector space <math>V</math> over the field <math>F</math> can be written as <math>\dim_F(V)</math> or as <math>[V : F],</math> read "dimension of <math>V</math> over <math>F</math>". When <math>F</math> can be inferred from context, <math>\dim(V)</math> is typically written.