How to detect redundant equation from a system of nonlinear equation?

[gohkgohkgohk]

How to detect redundant equation from a system of nonlinear equation?
It means how to find out a system of nonlinear equation is "linear independence"?
One equation from the system can not be represented by the others in the system of nonlinear equation.

 

[Eynstone]

Welcome gohkgohkgohk .
I'm afraid there's no general method of spotting a redundant equation from a set. The Jacobian helps a liitle if you have differentiable non-linear equations.

 

[kaige]

Hi Eynstone, could you share more details on how to determine the redundant equation using Jacobian? My suspection is even its Jacobian matrix has redundant rows, the non-linear equation system may not have redundant equations.

 

[algebrat]

I checked the section on functional dependence from Gerald Folland's Advanced Calculus.

There it says, if you consider on open set U, then

Jacobian is zero throughout U if and only if the functions have some (possibly various) functional dependence(s) throughout U.

So it's not enough for the Jacobian to vanish at a single point, calculus needs an open set to draw conclusions. To deal with single points, I think is part of the practicality of the subject algebraic geometry, which more or less deals with systems of polynomials, as far as I know, but looks like a very abstract subject.

 

[blue_raver22]

To detect redundant equations in a system of nonlinear equations, one can use the concept of linear independence. This means that the equations in the system are not multiples of each other and are not dependent on each other to find a solution. One way to determine this is by using row reduction techniques such as Gaussian elimination or LU decomposition. If the equations reduce to a lower triangular or diagonal form, it indicates that there are redundant equations in the system. Another method is to calculate the determinant of the coefficient matrix. If the determinant is equal to zero, it means that the equations are linearly dependent and there are redundant equations. Additionally, one can also use software tools such as MATLAB or Mathematica to solve the system and determine if there are any redundant equations. Overall, detecting redundant equations in a system of nonlinear equations is crucial for simplifying the system and finding the most efficient solution.