A linear system of equations over a field is a set of 
linear equations in 
variables (sometimes called "unknowns").
Linear systems can be represented in matrix form as the matrix
equation
 |
(1)
|
where 
is the matrix of coefficients, 
is the column vector of variables, and 
is the column vector of constants.
The system is consistent if and only if
![rank(A)=rank[A|b],](/images/equations/LinearSystemofEquations/NumberedEquation2.svg) |
(2)
|
where ![[A|b]](/images/equations/LinearSystemofEquations/Inline6.svg)
is the augmented matrix. Equivalently, its reduced
row-echelon form must not contain a row of the form
![[0,...,0|1]](/images/equations/LinearSystemofEquations/Inline7.svg)
.
When the system is consistent and 
is one particular solution, the complete solution set is
the affine coset
 |
(3)
|
It is therefore an affine space whose associated vector space is the null
space of 
.
Its dimension is 
(Shoup 2009).
If 
and 
is a nonsingular matrix, then the system has
a unique solution in the 
variables. In particular, as shown by Cramer's
rule, there is a unique solution if 
has a matrix inverse 
.
In this case,
 |
(4)
|
If 
,
then the system is homogeneous and 
is always a solution. It is the unique solution if and only
if the null space of 
is trivial.
If 
,
the system is overdetermined and is generally inconsistent. If 
, the system is underdetermined and, if consistent, has
free variables. Row reduction of the augmented
matrix identifies the pivot variables, free variables, and any inconsistency.
