Consistent and Inconsistent Linear Inhomogeneous Equations - tutorial 2 | applied electronics engineering

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Consistent and Inconsistent Linear Inhomogeneous Equations - tutorial 2

By Applied Electronics - Saturday, February 4, 2017 No Comments
Here we explain what is Consistent and Inconsistent Linear Inhomogeneous Equations. Matlab can be conviniently used to solve system of linear equations. But before showing how to do this in Matlab we need to know first the notions and theory behind system of linear equations. In linear algebra and in particular in system of linear equation we come across a notion of consistent and inconsistent inhomogeneous equations.

This is the 2nd part of tutorial Solving Linear Equations with Matlab

Equation of System of linear equation

The general form of notation of system of linear equations is,

Ax = b

where A is n by n square matrix, b is n by 1 column vector and x is also n by 1 column vector.

Homogeneous and Inhomogeneous System of Linear Equation

In the above equation, if b is Zero then the system of equation is called homogeneous and if b is non-Zero the system of equation is called inhomogeneous.

Solution to System of linear Equation

 Generally, if the system of linear equation gives some solution then the system of linear equations is said to be consistent otherwise the system of linear equation is said to be inconsistent. That is consistent means there is some solution and inconsistent means there is no solution. This notion is also applicable to non-linear systems.

Consistent Inhomogeneous System and Inconsistent Inhomogeneous System

  It can be proved that all homogeneous systems consistent therefore when we talk about consistent and inconsistent then we are talking about inhomogeneous systems. Hence, if the system of equations is inhomogeneous then we can have consistent or inconsistent system, that is consistent inhomogeneous system or inconsistent inhomogeneous system.

 Examples:

Consider inhomogeneous system of linear equation in 3 variables x1, x2 and x3. Physically each equation is a plane in space.

Consistent inhomogeneous system

 In this example, the three planes intersect at a point and the system of linear equations is said to consistent because there is a unique solution.

Consistent without unique solution

In this example, the three planes intersect in a line. So there is no unique point of intersection. All points on the line is solution to the system of linear equations. But still there is solution hence this is also an example of consistent inhomogeneous system of linear equations.
 


Inconsistent

In this example, the two planes S1 and S3 never meet. The three planes do not meet at any point and hence there is no solution and the system of linear equation describing these is said to be inconsistent.

Consistent and Inconsistent Linear Inhomogeneous Equations - tutorial 2
 Inconsistent

This is another example of inconsistent inhomogeneous system of linear equation. The planes have no common intersecting points.

Consistent and Inconsistent Linear Inhomogeneous Equations - tutorial 2

So this was about Consistent and Inconsistent Linear Inhomogeneous Equations. In the next tutorial we explain how to solve inhomogeneous equation system.

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