Editing Kernel (linear algebra) (section)

===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)]].