Row and column spaces

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, the column space (also called the range or <a href="/facts/Image_(mathematics)/KANefMAl">image</a>) of a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> A is the <a href="/facts/Linear_span/vPjclEtb">span</a> (set of all possible <a href="/facts/Linear_combination/CVjL6kuN">linear combinations</a>) of its <a href="/facts/Column_vector/pPuRr9Xk">column vectors</a>. The column space of a matrix is the <a href="/facts/Image_(mathematics)/KANefMAl">image</a> or <a href="/facts/Range_of_a_function/66CeTV6u">range</a> of the corresponding <a href="/facts/Matrix_transformation/qAwPXmpR">matrix transformation</a>.
Let 
 
 
 
 F
 
 
 {\displaystyle F}
 
 be a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>. The column space of an m × n matrix with components from 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is a <a href="/facts/Linear_subspace/2wtKTgHL">linear subspace</a> of the <a href="/facts/Examples_of_vector_spaces/tNTbVppn">m-space</a> 
 
 
 
 
 F
 
 m
 
 
 
 
 {\displaystyle F^{m}}
 
. The <a href="/facts/Dimension_(linear_algebra)/z8m02A3W">dimension</a> of the column space is called the <a href="/facts/Rank_(linear_algebra)/Yp9b5LK3">rank</a> of the matrix and is at most min(m, n). A definition for matrices over a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> 
 
 
 
 R
 
 
 {\displaystyle R}
 
 is also possible.
The row space is defined similarly.
The row space and the column space of a matrix A are sometimes denoted as C(AT) and C(A) respectively.
This article considers matrices of <a href="/facts/Real_number/R02gw5Pb">real numbers</a>. The row and column spaces are subspaces of the <a href="/facts/Real_coordinate_space/tEtvw03F">real spaces</a> 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 and 
 
 
 
 
 
 R
 
 
 m
 
 
 
 
 {\displaystyle \mathbb {R} ^{m}}
 
 respectively.

Row and column spaces open-in-new

Row and column spaces