Editing Rank (linear algebra) (section)

{{Short description|Dimension of the column space of a matrix}}
In [[linear algebra]], the '''rank''' of a [[matrix (mathematics)|matrix]] {{mvar|A}} is the [[Dimension (vector space)|dimension]] of the [[vector space]] generated (or [[Linear span|spanned]]) by its columns.<ref>{{Harvard citation text|Axler|2015}} pp. 111-112, §§ 3.115, 3.119</ref><ref name=":0">{{Harvard citation text|Roman|2005}} p. 48, § 1.16</ref><ref>Bourbaki, ''Algebra'', ch. II, §10.12, p. 359</ref> This corresponds to the maximal number of [[linearly independent]] columns of {{mvar|A}}. This, in turn, is identical to the dimension of the vector space spanned by its rows.<ref name="mackiw">{{Citation| last=Mackiw| first=G. | title=A Note on the Equality of the Column and Row Rank of a Matrix | year=1995| journal=[[Mathematics Magazine]] | volume=68| issue=4 | pages=285–286 | doi=10.1080/0025570X.1995.11996337 }}</ref> Rank is thus a measure of the "[[Degenerate form|nondegenerateness]]" of the [[system of linear equations]] and [[linear transformation]] encoded by {{mvar|A}}. There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics.

The rank is commonly denoted by {{math|rank(''A'')}} or {{math|rk(''A'')}};<ref name=":0" /> sometimes the parentheses are not written, as in {{math|rank ''A''}}.<ref group="lower-roman">Alternative notation includes <math>\rho (\Phi)</math> from {{Harvard citation text|Katznelson|Katznelson|2008|p=52, §2.5.1}} and {{Harvard citation text|Halmos|1974|p=90, § 50}}.</ref>