Editing Row and column spaces (section)

==Overview==
Let {{mvar|A}} be an {{mvar|m}}-by-{{mvar|n}} matrix. Then
* {{math|1=rank(''A'') = dim(rowsp(''A'')) = dim(colsp(''A''))}},<ref>{{harvtxt|Anton|1987|p=183}}</ref>
* {{math|rank(''A'')}} = number of [[Pivot element|pivots]] in any echelon form of {{mvar|A}},
* {{math|rank(''A'')}} = the maximum number of linearly independent rows or columns of {{mvar|A}}.<ref>{{harvtxt|Beauregard|Fraleigh|1973|p=254}}</ref>

If the matrix represents a [[linear transformation]], the column space of the matrix equals the [[image (mathematics)|image]] of this linear transformation.

The column space of a matrix {{mvar|A}} is the set of all linear combinations of the columns in {{mvar|A}}.  If {{math|1=''A'' = ['''a'''<sub>1</sub> ⋯ '''a'''<sub>''n''</sub>]}}, then {{math|1=colsp(''A'') = span({{mset|'''a'''<sub>1</sub>, ..., '''a'''<sub>''n''</sub>}})}}.

Given a matrix {{mvar|A}}, the action of the matrix {{mvar|A}} on a vector {{math|'''x'''}} returns a linear combination of the columns of {{mvar|A}} with the coordinates of {{math|'''x'''}} as coefficients; that is, the columns of the matrix generate the column space.

===Example===
Given a matrix {{mvar|J}}:
:<math>
  J =
  \begin{bmatrix}
    2 & 4 & 1 & 3 & 2\\
    -1 & -2 & 1 & 0 & 5\\
    1 & 6 & 2 & 2 & 2\\
    3 & 6 & 2 & 5 & 1
  \end{bmatrix}
</math>
the rows are
<math>\mathbf{r}_1 = \begin{bmatrix} 2 & 4 & 1 & 3 & 2 \end{bmatrix}</math>,
<math>\mathbf{r}_2 = \begin{bmatrix} -1 & -2 & 1 & 0 & 5 \end{bmatrix}</math>,
<math>\mathbf{r}_3 = \begin{bmatrix} 1 & 6 & 2 & 2 & 2 \end{bmatrix}</math>,
<math>\mathbf{r}_4 = \begin{bmatrix} 3 & 6 & 2 & 5 & 1 \end{bmatrix}</math>.
Consequently, the row space of {{mvar|J}} is the subspace of <math>\R^5</math> [[linear span|spanned]] by {{math|{{mset| '''r'''<sub>1</sub>, '''r'''<sub>2</sub>, '''r'''<sub>3</sub>, '''r'''<sub>4</sub> }}}}.   
Since these four row vectors are [[Linear independence|linearly independent]], the row space is 4-dimensional.  Moreover, in this case it can be seen that they are all [[orthogonality|orthogonal]] to the vector {{math|1='''n''' = [6, −1, 4, −4, 0]}} ({{math|1='''n'''}} is an element of the [[Kernel (linear algebra)|kernel]] of {{mvar|J}} ), so it can be deduced that the row space consists of all vectors in <math>\R^5</math> that are orthogonal to {{math|'''n'''}}.