The Co-efficients, Variables, and the Constants of the System of Linear Equations given above can be represented using Matrices as given in the following
The Matrix A is called the Co-efficient Matrix. The Matrix/Vector X is called the Variable Matrix/Vector. The Matrix/Vector K is called the Constant Matrix/Vector.
Using Matrix A,X and K the System of Linear Equations can be represented using the following Matrix Equation
Based on Number of Equations M and Number of Variables N, any System of Linear Equations can be of One of the following 3 Types
System of Linear Equations having Number of Equations Greater than the Number of Variables (i.e. M>N).
System of Linear Equations having Number of Equations Equal to the Number of Variables (i.e. M=N).
System of Linear Equations having Number of Equations Lesser than the Number of Variables (i.e. M<N).
Based on Value of the Constants of Linear Equations, any System of Linear Equations can be of One of the following 2 Types
Homogeneous System of Linear Equations: If the Constants of the All the Linear Equations (i.e. k1,k2,...,km) are 0 then such a System of Linear Equations is called
Homogeneous System of Linear Equations. These System of Linear Equations can either have only a Single Trivial Solution (i.e. value of all Variables (x1,x2,...,xn equal to 0) , or can have an
Infinite Number of Non-Trivial Solutions. These Non-Trivial Solutions are themselves represented by Linear Equations of Variables (x1,x2,...,xn).
Solutions of the Homogeneous System of Linear Equations is same as the NULL Space of the Coefficient Matrix.
Non-Homogeneous System of Linear Equations: If Constant of at least One of the Linear Equations (i.e. k1,k2,...,km) is Non-Zero, then such a System of Linear Equations is called
Non-Homogeneous System of Linear Equations. These System of Linear Equations can either have No Solution, a Unique Solution or Infinitely Many Solutions.
Non-Homogeneous System of Linear Equations with Unique Solution or Infinitely Many Solutions are also called Consistent.
Non-Homogeneous System of Linear Equations with No Solution are also called Inconsistent.
Also, finding solutions of the Non-Homogeneous System of Linear Equations is same as Finding whether the Constant Vector/Matrix belongs to the same Column Space as the Coefficient Matrix.
If One or More Solutions Exist (i.e. for Consistent System of Linear Equations), then the Constant Vector/Matrix belongs to the same Column Space as the Coefficient Matrix.
If Solution Does Not Exist (i.e. for Non-Consistent System of Linear Equations), then the Constant Vector/Matrix Does Not belong to the same Column Space as the Coefficient Matrix.
The following methods are generally used to Find Solutions for System of Linear Equations