Editing Row and column spaces (section)

===Definition===
Let {{mvar|K}} be a [[field (mathematics)|field]] of [[scalar (mathematics)|scalars]]. Let {{mvar|A}} be an {{math|''m'' × ''n''}} matrix, with row vectors {{math|'''r'''<sub>1</sub>, '''r'''<sub>2</sub>, ..., '''r'''<sub>''m''</sub>}}.  A [[linear combination]] of these vectors is any vector of the form
:<math>c_1 \mathbf{r}_1 + c_2 \mathbf{r}_2 + \cdots + c_m \mathbf{r}_m,</math>
where {{math|''c''<sub>1</sub>, ''c''<sub>2</sub>, ..., ''c<sub>m</sub>''}} are scalars.  The set of all possible linear combinations of {{math|'''r'''<sub>1</sub>, ..., '''r'''<sub>''m''</sub>}} is called the '''row space''' of {{mvar|A}}.  That is, the row space of {{mvar|A}} is the [[linear span|span]] of the vectors {{math|'''r'''<sub>1</sub>, ..., '''r'''<sub>''m''</sub>}}.

For example, if
:<math>A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 0 \end{bmatrix},</math>
then the row vectors are {{math|1='''r'''<sub>1</sub> = [1, 0, 2]}} and {{math|1='''r'''<sub>2</sub> = [0, 1, 0]}}.  A linear combination of {{math|'''r'''<sub>1</sub>}} and {{math|'''r'''<sub>2</sub>}} is any vector of the form
:<math>c_1 \begin{bmatrix}1 & 0 & 2\end{bmatrix} + c_2 \begin{bmatrix}0 & 1 & 0\end{bmatrix} = \begin{bmatrix}c_1 & c_2 & 2c_1\end{bmatrix}.</math>
The set of all such vectors is the row space of {{mvar|A}}.  In this case, the row space is precisely the set of vectors {{math|(''x'', ''y'', ''z'') ∈ ''K''<sup>3</sup>}} satisfying the equation {{math|1=''z'' = 2''x''}} (using [[Cartesian coordinates]], this set is a [[plane (mathematics)|plane]] through the origin in [[three-dimensional space]]).

For a matrix that represents a homogeneous [[system of linear equations]], the row space consists of all linear equations that follow from those in the system.

The column space of {{mvar|A}} is equal to the row space of {{math|''A''<sup>T</sup>}}.