Editing Kernel (linear algebra) (section)

==Representation as matrix multiplication==
Consider a linear map represented as a {{math|''m'' × ''n''}} matrix {{mvar|A}} with coefficients in a [[field (mathematics)|field]] {{mvar|K}} (typically <math>\mathbb{R}</math> or <math>\mathbb{C}</math>), that is operating on column vectors {{math|'''x'''}} with {{mvar|n}} components over {{mvar|K}}.
The kernel of this linear map is the set of solutions to the equation {{math|1=''A'''''x''' = '''0'''}}, where {{math|'''0'''}} is understood as the [[zero vector]]. The [[dimension (vector space)|dimension]] of the kernel of ''A'' is called the '''nullity''' of ''A''. In [[set-builder notation]],
<math display="block">\operatorname{N}(A) = \operatorname{Null}(A) = \operatorname{ker}(A) = \left\{ \mathbf{x}\in K^n \mid A\mathbf{x} = \mathbf{0} \right\}.</math>
The matrix equation is equivalent to a homogeneous [[system of linear equations]]:
<math display="block">A\mathbf{x}=\mathbf{0} \;\;\Leftrightarrow\;\;
\begin{alignat}{7}
a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \;\cdots\; + \;&& a_{1n} x_n &&\; = \;&&& 0      \\
a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \;\cdots\; + \;&& a_{2n} x_n &&\; = \;&&& 0      \\
           &&       &&            &&                    &&            &&\vdots\ \;&&&    \\
a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \;\cdots\; + \;&& a_{mn} x_n &&\; = \;&&& 0\text{.}      \\
\end{alignat}</math>
Thus the kernel of ''A'' is the same as the solution set to the above homogeneous equations.

===Subspace properties===
The kernel of a {{math|''m'' × ''n''}} matrix {{mvar|A}} over a field {{mvar|K}} is a [[linear subspace]] of {{math|'''K'''<sup>''n''</sup>}}. That is, the kernel of {{mvar|A}}, the set {{math|Null(''A'')}}, has the following three properties:
# {{math|Null(''A'')}} always contains the [[zero vector]], since {{math|1=''A'''''0''' = '''0'''}}.
# If {{math|'''x''' ∈ Null(''A'')}} and {{math|'''y''' ∈ Null(''A'')}}, then {{math|'''x''' + '''y''' ∈ Null(''A'')}}. This follows from the distributivity of matrix multiplication over addition.
# If {{math|'''x''' ∈ Null(''A'')}} and {{mvar|c}} is a [[scalar (mathematics)|scalar]] {{math|''c'' ∈ ''K''}}, then {{math|''c'''''x''' ∈ Null(''A'')}}, since {{math|1=''A''(''c'''''x''') = ''c''(''A'''''x''') = ''c'''''0''' = '''0'''}}.

===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>

===Left null space===
The '''left null space''', or [[cokernel]], of a matrix {{mvar|A}} consists of all column vectors {{math|'''x'''}} such that {{math|1='''x'''<sup>T</sup>''A'' = '''0'''<sup>T</sup>}}, where T denotes the [[transpose]] of a matrix.  The left null space of {{mvar|A}} is the same as the kernel of {{math|''A''<sup>T</sup>}}. The left null space of {{mvar|A}} is the orthogonal complement to the [[column space]] of {{mvar|A}}, and is dual to the [[cokernel]] of the associated linear transformation. The kernel, the row space, the column space, and the left null space of {{mvar|A}} are the '''four fundamental subspaces''' associated with the matrix {{mvar|A}}.

===Nonhomogeneous systems of linear equations===
The kernel also plays a role in the solution to a nonhomogeneous system of linear equations:
<math display="block">A\mathbf{x} = \mathbf{b}\quad \text{or} \quad \begin{alignat}{7}
a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \;\cdots\; + \;&& a_{1n} x_n &&\; = \;&&& b_1      \\
a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \;\cdots\; + \;&& a_{2n} x_n &&\; = \;&&& b_2      \\
           &&       &&            &&                    &&            &&\vdots\ \;&&&       \\
a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \;\cdots\; + \;&& a_{mn} x_n &&\; = \;&&& b_m      \\
\end{alignat}</math>
If {{math|'''u'''}} and {{math|'''v'''}} are two possible solutions to the above equation, then
<math display="block">A(\mathbf{u} - \mathbf{v}) = A\mathbf{u} - A\mathbf{v} = \mathbf{b} - \mathbf{b} = \mathbf{0}</math>
Thus, the difference of any two solutions to the equation {{math|1=''A'''''x''' = '''b'''}} lies in the kernel of {{mvar|A}}.

It follows that any solution to the equation {{math|1=''A'''''x''' = '''b'''}} can be expressed as the sum of a fixed solution {{math|'''v'''}} and an arbitrary element of the kernel. That is, the solution set to the equation {{math|1=''A'''''x''' = '''b'''}} is
<math display="block">\left\{ \mathbf{v}+\mathbf{x} \mid A \mathbf{v}=\mathbf{b} \land \mathbf{x}\in\operatorname{Null}(A) \right\},</math>
Geometrically, this says that the solution set to {{math|1=''A'''''x''' = '''b'''}} is the [[translation (geometry)|translation]] of the kernel of {{mvar|A}} by the vector {{math|'''v'''}}. See also [[Fredholm alternative]] and [[flat (geometry)]].