A linear system of equations 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 solutions.
If 
,
then the system is (in general) overdetermined and there is no solution.
If 
and the matrix 
is nonsingular, 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,
 |
(2)
|
If 
,
then the solution is simply 
. If 
has no matrix inverse, then
the solution set is the translate of a subspace of dimension
less than 
or the empty set.
If two equations are multiples of each other, solutions are of the form
 |
(3)
|
for 
a real number. More generally, if 
, then the system is underdetermined. In this case, elementary row and column operations
can be used to solve the system as far as possible, then the first 
components can be solved in terms of the last 
components to find the solution space.
