Linear System (System of linear equations)

It’s a collection of one ore more linear equations involving the same variables, like:

A system could be:

Determinated if there are the same number of equation unknowns
Overdeterminated if there are more equation that the number of unknowns
Underdeterminated if the number of equation is fewer than the number of unknowns

Solutions

A solution to a linear system is an assignement of values is an assignement to variables such that all the equations are simultaneously satisfied.

Example of solution in a linear system with two equations: LinearSystemSolutions

A general system of m linear equations with n unknown and coefficients can be written as:

Which can be written in vectoral or matrix form.

A system of equations is called homogeneous if all the costant terms are zero:

The solution is called trivial solution. Any solution with atleast a non zero is called nontrivial solution.

The general form can be seen as a matrix equation: , with the matrix A being of size representing coefficients, and the vector b are the solutions.