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System of linear equations

In mathematics and linear algebra, a system of linear equations is a set of linear equations such as

3 x1 + 2 x2 - x3 = 1
2 x1 - 2 x2 + 4x3 = -2
-x1 + 1/2 x2 - x3 = 0

The problem is to find those values for the unknowns x1, x2 and x3 which satisfy all three equations simultaneously.

Systems of linear equations belong to the oldest problems in mathematics and they have many applications, such as in digital signal processing, estimation, forecasting and generally in linear programming and in the approximation of non-linear problems in numerical analysis. An efficient way to solve systems of linear equations is given by the Gauss-Jordan elimination or by the Cholesky decomposition.

In general, a system with m linear equations and n unknowns can be written as

a11x1 + a12x2 + … + a1nxn = b1
a21x1 + a22x2 + … + a2nxn = b2
am1x1 + am2x2 + … + amnxn = bm

where x1,...,xn are the unknowns and the numbers aij are the coefficients of the system. We write the coefficients in the matrix in the following way

This can also be represented as
A x = b,
where A is an m-by-n matrix above, x is a column vector with n entries and b is a column vector with m entries. The above mentioned Gauss-Jordan elimination applies to all these systems, even if the coefficients come from an arbitrary field.

If the field is infinite (as in the case of the real or complex numbers), then only the following three cases are possible for any given system of linear equations:

A system of the form
A x = 0
is called a homogenous system of linear equations. The set of all solutions of such a homogeneous system is always a linear subspace of F n, where F is the field of coefficients.

Especially in view of the abiove applications, several more efficient alternatives to Gauss-Jordan elimination have been developed for a wide diversity of special cases. Many of these improved algorithms are of complexity O(n²). Some of the most common special cases are: