Editing Row and column spaces (section)

===Relation to the left null space===
The [[left null space]] of {{mvar|A}} is the set of all vectors {{math|'''x'''}} such that {{math|1='''x'''<sup>T</sup>''A'' = '''0'''<sup>T</sup>}}.  It is the same as the [[kernel (matrix)|null space]] of the [[transpose]] of {{mvar|A}}. The product of the matrix {{math|''A''<sup>T</sup>}} and the vector {{math|'''x'''}} can be written in terms of the [[dot product]] of vectors:
:<math>A^\mathsf{T}\mathbf{x} = \begin{bmatrix} \mathbf{v}_1 \cdot \mathbf{x} \\ \mathbf{v}_2 \cdot \mathbf{x} \\ \vdots \\ \mathbf{v}_n \cdot \mathbf{x} \end{bmatrix},</math>
because [[row vector]]s of {{math|''A''<sup>T</sup>}} are transposes of column vectors {{math|'''v'''<sub>''k''</sub>}} of {{mvar|A}}.  Thus {{math|1=''A''<sup>T</sup>'''x''' = '''0'''}} if and only if {{math|'''x'''}} is [[orthogonal]] (perpendicular) to each of the column vectors of {{mvar|A}}.

It follows that the left null space (the null space of {{math|''A''<sup>T</sup>}}) is the [[orthogonal complement]] to the column space of {{mvar|A}}.

For a matrix {{mvar|A}}, the column space, row space, null space, and left null space are sometimes referred to as the ''four fundamental subspaces''.