section_path: Linear systems
section_name: Linear systems
paragraph_text: 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.