Linear systems: zero, one, infinite solutions

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Discussion Overview

The discussion revolves around the conditions under which a system of linear equations can have no solution, one unique solution, or infinitely many solutions. Participants explore the implications of linear versus nonlinear systems and the potential for mixed systems containing both types of equations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that a system of linear equations can yield either no solution, one solution, or infinitely many solutions based on the geometric interpretation of lines in a plane.
  • Others argue that the nature of the solutions depends on the number of equations relative to the number of unknowns, with cases presented for under-determined, fully determined, and overdetermined systems.
  • A participant questions whether the original inquiry pertains specifically to linear systems or if it encompasses broader scenarios, including nonlinear systems.
  • There is a suggestion that nonlinear systems can indeed have a finite number of solutions, which contrasts with the behavior of linear systems.
  • Some participants note that a mixed system containing both linear and nonlinear equations would be classified as nonlinear, affecting the methods used for finding solutions.
  • It is mentioned that while nonlinear equations can be organized in a matrix, linear algebra techniques cannot be applied to solve them directly.

Areas of Agreement / Disagreement

Participants generally agree on the basic classifications of solutions for linear systems but express differing views on the implications of including nonlinear equations and the nature of mixed systems. The discussion remains unresolved regarding the broader implications of nonlinear systems and their solutions.

Contextual Notes

Limitations include the dependence on the definitions of under-determined, fully determined, and overdetermined systems, as well as the unresolved nature of how mixed systems behave compared to purely linear or nonlinear systems.

fisico30
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Hello forum,
why does a system of linear equations really have no solution or one unique solution, or infinite solutions?

What forbids a system to a finite number of solutions?

thanks
fisico30
 
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Consider two lines in a plane. Either the lines intersect at a point, are parallel but don't overlap, or are parallel and coincident. 1, 0, infinitely many solutions. There are no other possibilities. Any "curving" that would make multiple finite solutions possible would be nonlinear.
 
That's true for two lines in a plane but is that what the OP was asking?

Consider the following three systems of linear equations

1) ax+by+cz=0; dx+ey+fz=0

2) ax+by+cz=0; dx+ey+fz=0; gx+hy+iz=0

3) ax+by+cz=0; dx+ey+fz=0; gx+hy+iz=0; jx+ky+lz=0

What then?
 
All these planes go through the origin? In that case, only in case (1) is a single point solution forbidden. All others could, in principle, admit solutions that are either a single point, a line, or a plane. None of those could have no solution.
 
Ok,,
thanks everyone.
So if the system of composed of linear equations, then I can see how, 1,0, infinity would the the solutions...
The linear equations can be algebraic or differential, correct? The same solutions (1,0, infinity) would work...

If the system was made of nonlinear equations, then there could be a finite number of solutions, correct?

Is it possible to have a mixed, hybrid, system, composed of linear equations and nonlinear equations?

thanks
fisico30
 
Hello fisico,

Yes you can have a mixture of linear and non linear equations, but the system is automatically non linear if it includes even one non linear equation.

Please note that my examples are different from Murphrid ( who is not wrong) because they show a different situation.

Case (1) is an under-determined system because there are more unknowns than equations.

Case (2) is fully determined since the number of equations matches the number of unknowns.

Case (3) is overdetermined since there are more equations than unknowns.
 
Thanks Studiot,

I guess if the system is nonlinear (at least one nonlinear equation), then we cannot even think about matrices and linear algebra to find the solution(s), correct?

Matrices are only useful for linear systems...

In nonlinear systems, a consistent system can have more than 1 unique solution...
thanks
fisico30
 
You can assemble non linear equations within a matrix, you just can't use linear matrix algebra to solve the system.

Normally a great deal of effort goes into find linear approximations or substitutions or restricted ranges over which linearity can be assumed, in order to use matrix algorithms. It all depends upon the equations.
 

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