PREMISES AND QUESTION
Question: is there rigorous and general way to define the idea of “independent/dependent system” - and by it I mean a system that has no redundant equations?
I often found the notion of “independent set of equations”, but I always find it applied to linear equations (thus to linear system) - for example in linear algebra, in applying kirchoff’s laws to solve a circuit etc. It seems to me that it would be useful to generalize (rigorously) this notion to the cases where I can have all types of equations, since being able to avoid putting a lot of redundant equations in a system to solve a problem seems useful.
- Note that I’m not really even sure that we can generalize the idea of equation (so just one definition that includes algebraic, trascendental etc.), because I only know a rigorous definition of algebraic equations, which is a formula $s(x)=t(x)$ where $s$ and $t$ are rational expressions. Because of this, I obviously accept answers that are able to generalize independent/dependent systems even only to a certain degree (for example if one tells me a definition is good for all systems with both trascendental or algebraic equations, or anything). Or am I trying to ask something that cannot really be generalized because it’s too vague?
MY TRY
I had an idea for a recursive definition:
- A system of $1$ equation is independent iff the equation is not $0=0$.
- A system of $n$ equations is said to be “dependent” if and only if the solutions of the $S’$ system made by the first $n-1$ equations is the same as the set of solutions of $S$.
SIMPLE CASE THAT I ALREADY KNOW OF: ALGEBRAIC LINEAR SYSTEMS
One example: with an algebraic linear system $S$ (made only by algebraic linear equations), we could just say, by using the gauss elimination algorithm, that $S$ is independent if and only if we don’t have a $0=0$ equation when reducing it to row echelon form. But in non algebraic systems, where sometimes we cannot isolate the variables explicitly (for example with an equation of type $e^x + x=0$) we could not be able to get a $0=0$ equation, even though maybe the equations are redundant.
This is why I’m not sure if my general definition is still too particular (it excludes some other cases or type of equations that I didn’t think of) or too generic (it includes some cases that I should not want to include).
A candidate definition could be "a system is independent if removing any of the equations changes the solution set".