Editing Kernel (linear algebra) (section)

===The row space of a matrix===
{{main|Rank–nullity theorem}}
The product ''A'''''x''' can be written in terms of the [[dot product]] of vectors as follows:
<math display="block">A\mathbf{x} = \begin{bmatrix} \mathbf{a}_1 \cdot \mathbf{x} \\ \mathbf{a}_2 \cdot \mathbf{x} \\ \vdots \\ \mathbf{a}_m \cdot \mathbf{x} \end{bmatrix}.</math>

Here, {{math|'''a'''<sub>1</sub>, ... , '''a'''<sub>''m''</sub>}} denote the rows of the matrix {{mvar|A}}. It follows that {{math|'''x'''}} is in the kernel of {{mvar|A}}, if and only if {{math|'''x'''}} is [[orthogonality|orthogonal]] (or perpendicular) to each of the row vectors of {{mvar|A}} (since orthogonality is defined as having a dot product of 0).

The [[row space]], or coimage, of a matrix {{mvar|A}} is the [[linear span|span]] of the row vectors of {{mvar|A}}. By the above reasoning, the kernel of {{mvar|A}} is the [[orthogonal complement]] to the row space. That is, a vector {{math|'''x'''}} lies in the kernel of {{mvar|A}}, if and only if it is perpendicular to every vector in the row space of {{mvar|A}}.

The dimension of the row space of {{mvar|A}} is called the [[rank (linear algebra)|rank]] of ''A'', and the dimension of the kernel of {{mvar|A}} is called the '''nullity''' of {{mvar|A}}. These quantities are related by the [[rank–nullity theorem]]<ref name=":1" />
<math display="block">\operatorname{rank}(A) + \operatorname{nullity}(A) = n.</math>