Editing System of linear equations (section)

==Homogeneous systems==
{{See also|Homogeneous differential equation}}

A system of linear equations is '''homogeneous''' if all of the constant terms are zero:
: <math>\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. \\
\end{alignat}</math>
A homogeneous system is equivalent to a matrix equation of the form
: <math>A\mathbf{x}=\mathbf{0}</math>
where ''A'' is an {{nowrap|''m'' × ''n''}} matrix, '''x''' is a column vector with ''n'' entries, and '''0''' is the [[zero vector]] with ''m'' entries.

===Homogeneous solution set===
Every homogeneous system has at least one solution, known as the ''zero'' (or ''trivial'') solution, which is obtained by assigning the value of zero to each of the variables. If the system has a [[non-singular matrix]] ({{math|det(''A'') ≠ 0}}) then it is also the only solution.  If the system has a singular matrix then there is a solution set with an infinite number of solutions.  This solution set has the following additional properties:
# If '''u''' and '''v''' are two [[vector (mathematics)|vectors]] representing solutions to a homogeneous system, then the vector sum {{nowrap|'''u''' + '''v'''}} is also a solution to the system.
# If '''u''' is a vector representing a solution to a homogeneous system, and ''r'' is any [[scalar (mathematics)|scalar]], then ''r'''''u''' is also a solution to the system.
These are exactly the properties required for the solution set to be a [[linear subspace]] of '''R'''<sup>''n''</sup>. In particular, the solution set to a homogeneous system is the same as the [[Kernel (matrix)|null space]] of the corresponding matrix ''A''.

===Relation to nonhomogeneous systems===
There is a close relationship between the solutions to a linear system and the solutions to the corresponding homogeneous system:
: <math>A\mathbf{x}=\mathbf{b}\qquad \text{and}\qquad A\mathbf{x}=\mathbf{0}.</math>
Specifically, if '''p''' is any specific solution to the linear system {{nowrap|1=''A'''''x''' = '''b'''}}, then the entire solution set can be described as
: <math>\left\{ \mathbf{p}+\mathbf{v} : \mathbf{v}\text{ is any solution to }A\mathbf{x}=\mathbf{0} \right\}.</math>
Geometrically, this says that the solution set for {{nowrap|1=''A'''''x''' = '''b'''}} is a [[translation (geometry)|translation]] of the solution set for {{nowrap|1=''A'''''x''' = '''0'''}}. Specifically, the [[flat (geometry)|flat]] for the first system can be obtained by translating the [[Euclidean subspace|linear subspace]] for the homogeneous system by the vector '''p'''.

This reasoning only applies if the system {{nowrap|1=''A'''''x''' = '''b'''}} has at least one solution. This occurs if and only if the vector '''b''' lies in the [[image (mathematics)|image]] of the [[linear transformation]] ''A''.