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

: 3 x 1 + 2 x 2 − x 3 = 1 : 2 x 1 − 2 x 2 + 4 x 3 = −2 : − x 1 + ½ x 2 − x 3 = 0.

The problem is to find those values for the unknowns x 1, x 2 and x 3 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 : a 11 x 1 + a 12 x 2 + … + a 1 n x n = b 1 : a 21 x 1 + a 22 x 2 + … + a 2 n x n = b 2 :     : :     : : a m 1 x 1 + a m 2 x 2 + … + a mn x n = b m ,

where x 1, ... , x n are the unknowns and the numbers a ij are the coefficients of the system. We can separate the coefficients in a matrix as follows: : egin{bmatrix} a_{11} & a_{12} & cdots & a_{1n} \ a_{21} & a_{22} & cdots & a_{2n} \ vdots & vdots & ddots & vdots \ a_{m1} & a_{m2} & cdots & a_{mn} end{bmatrix}

egin{bmatrix} x_1 \ x_2 \ vdots \ x_n end{bmatrix} = egin{bmatrix} b_1 \ b_2 \ vdots \ b_m end{bmatrix}

If we represent each matrix by a single letter, this becomes : A x = b, where A is an m -by- n matrix (mathematics) 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 (mathematics).

If the field is infinite (as in the case of the real number 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 called the null space of the matrix A, it is written as Nul A .

Especially in view of the above 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 Big O notation. Some of the most common special cases are:

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