structural:expansion
Linear systems

A finite set of linear equations in a finite set of variables, for example, x, x,..., x, or x, y,..., z is called a system of linear equations or a linear system.

Systems of linear equations form a fundamental part of linear algebra. Historically, linear algebra and matrix theory has been developed for solving such systems. In the modern presentation of linear algebra through vector spaces and matrices, many problems may be interpreted in terms of linear systems.

For example, let

be a linear system.

To such a system, one may associate its matrix

and its right member vector

Let T be the linear transformation associated to the matrix M. A solution of the system (S) is a vector

such that

that is an element of the preimage of v by T.

Let (S′) be the associated homogeneous system, where the right-hand sides of the equations are put to zero:

The solutions of (S′) are exactly the elements of the kernel of T or, equivalently, M.

The Gaussian-elimination consists of performing elementary row operations on the augmented matrix

for putting it in reduced row echelon form. These row operations do not change the set of solutions of the system of equations. In the example, the reduced echelon form is

showing that the system (S) has the unique solution

It follows from this matrix interpretation of linear systems that the same methods can be applied for solving linear systems and for many operations on matrices and linear transformations, which include the computation of the ranks, kernels, matrix inverses.