Editing Row and column spaces (section)

{{Short description|Vector spaces associated to a matrix}}
[[File:Matrix Rows.svg|thumb|right|The row vectors of a [[matrix (mathematics)|matrix]]. The row space of this matrix is the vector space spanned by the row vectors.]]

[[Image:Matrix Columns.svg|thumb|right|The column vectors of a [[matrix (mathematics)|matrix]]. The column space of this matrix is the vector space spanned by the column vectors.]]

In [[linear algebra]], the '''column space''' (also called the '''range''' or [[Image (mathematics)|'''image''']]) of a [[matrix (mathematics)|matrix]] ''A'' is the [[Linear span|span]] (set of all possible [[linear combination]]s) of its [[column vector]]s. The column space of a matrix is the [[image (mathematics)|image]] or [[range of a function|range]] of the corresponding [[matrix transformation]].

Let <math>F</math> be a [[field (mathematics)|field]]. The column space of an {{math|''m'' × ''n''}} matrix with components from <math>F</math> is a [[linear subspace]] of the [[Examples of vector spaces#Coordinate space|''m''-space]] <math>F^m</math>.  The [[dimension (linear algebra)|dimension]] of the column space is called the [[rank (linear algebra)|rank]] of the matrix and is at most {{math|min(''m'', ''n'')}}.<ref name="ReferenceA">Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in Lay 2005, Meyer 2001, and Strang 2005.</ref> A definition for matrices over a [[ring (mathematics)|ring]] <math>R</math> [[#For matrices over a ring|is also possible]].

The '''row space''' is defined similarly.

The row space and the column space of a matrix {{mvar|A}} are sometimes denoted as {{math|'''''C'''''(''A''<sup>T</sup>)}} and {{math|'''''C'''''(''A'')}} respectively.<ref>{{Cite book|last=Strang|first=Gilbert|url=https://www.worldcat.org/oclc/956503593|title=Introduction to linear algebra|publisher=Wellesley-Cambridge Press|year=2016|isbn=978-0-9802327-7-6|edition=Fifth|location=Wellesley, MA|pages=128,168|oclc=956503593}}</ref>

This article considers matrices of [[real number]]s.  The row and column spaces are subspaces of the [[real coordinate space|real spaces]] <math>\R^n</math> and <math>\R^m</math> respectively.<ref>{{harvtxt|Anton|1987|p=179}}</ref>