Editing Row and column spaces (section)

===Relation to the null space===
The [[null space]] of matrix {{mvar|A}} is the set of all vectors {{math|'''x'''}} for which {{math|1=''A'''''x''' = '''0'''}}.  The product of the matrix {{mvar|A}} and the vector {{math|'''x'''}} can be written in terms of the [[dot product]] of vectors:
:<math>A\mathbf{x} = \begin{bmatrix} \mathbf{r}_1 \cdot \mathbf{x} \\ \mathbf{r}_2 \cdot \mathbf{x} \\ \vdots \\ \mathbf{r}_m \cdot \mathbf{x} \end{bmatrix},</math>
where {{math|'''r'''<sub>1</sub>, ..., '''r'''<sub>''m''</sub>}} are the row vectors of {{mvar|A}}.  Thus {{math|1=''A'''''x''' = '''0'''}} if and only if {{math|'''x'''}} is [[orthogonal]] (perpendicular) to each of the row vectors of {{mvar|A}}.

It follows that the null space of {{mvar|A}} is the [[orthogonal complement]] to the row space.  For example, if the row space is a plane through the origin in three dimensions, then the null space will be the perpendicular line through the origin.  This provides a proof of the [[rank–nullity theorem]] (see [[#Dimension|dimension]] above).

The row space and null space are two of the [[four fundamental subspaces]] associated with a matrix {{mvar|A}} (the other two being the [[column space]] and [[left null space]]).