Linear systems: zero, one, infinite solutions

[fisico30]

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

 

[Muphrid]

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.

 

[Studiot]

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?

 

[Muphrid]

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.

 

[fisico30]

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

 

[Studiot]

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.

 

[fisico30]

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

 

[Studiot]

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.