Editing Rank (linear algebra) (section)

== Main definitions ==

In this section, we give some definitions of the rank of a matrix.  Many definitions are possible; see [[#Alternative definitions|Alternative definitions]] for several of these.

The '''column rank''' of {{mvar|A}} is the [[dimension (linear algebra)|dimension]] of the [[column space]] of {{mvar|A}}, while the '''row rank''' of {{mvar|A}} is the dimension of the [[row space]] of {{mvar|A}}.

A fundamental result in linear algebra is that the column rank and the row rank are always equal. (Three proofs of this result are given in {{slink||2=Proofs that column rank = row rank}}, below.)  This number (i.e., the number of linearly independent rows or columns) is simply called the '''rank''' of {{mvar|A}}.

A matrix is said to have '''full rank''' if its rank equals the largest possible for a matrix of the same dimensions, which is the lesser of the number of rows and columns.  A matrix is said to be '''rank-deficient''' if it does not have full rank. The '''rank deficiency''' of a matrix is the difference between the lesser of the number of rows and columns, and the rank.

The rank of a [[linear map]] or operator <math>\Phi</math> is defined as the dimension of its [[Image (mathematics)|image]]:<ref>{{Harvard citation text|Hefferon|2020}} p. 200, ch. 3, Definition 2.1</ref><ref>{{Harvard citation text|Katznelson|Katznelson|2008}} p. 52, § 2.5.1</ref><ref>{{Harvard citation text|Valenza|1993}} p. 71, § 4.3</ref><ref>{{Harvard citation text|Halmos|1974}} p. 90, § 50</ref><math display="block">\operatorname{rank} (\Phi) := \dim (\operatorname{img} (\Phi))</math>where <math>\dim</math> is the dimension of a vector space, and <math>\operatorname{img}</math> is the image of a map.