How to know if an infinite system of linear equations has a solution

[f22rumaj]

Hello, I'm trying to solve a differential equation with boundary conditions which leads me to an infinite system of linear equations.
I can obtain an approximate solution of the problem just by considering only the first n terms so I have a system of n equations with n unknowns.
I've been trying to find an exact solution of the problem by other methods but I always failed, and I suspect that the boundary conditions are incompatible so the exact solution does not exist and that's what I'm trying to prove now.
I noticed that the condition number of the system (computationally calculated of course) increases as n increases and that could be a prove that when n is infinite so is the condition number and therefore the matrix is singular.
However I don't know how to prove this mathematically, I only can calculate the variation of the contidion number with n for n from 5 to 300 or maybe 400, no more.

Does anyone have some ideas to help me?

Thank you so much in advance!

 

[chiro]

Hey f22rumaj and welcome to the forums.

I don't know if this suggestion will help, but one thing that springs to mind is to use matrix norms to see if a solution exists or whether the matrix is 'stable' enough to give a solution.

We use this technique in the finite-case for getting a sense of how 'stable' a particular system in terms of getting a solution using numerical methods like Gauss-Jordan and Gauss-Siedel.

Now in operator algebras, we actually study norms for these operators which are essentially infinitexinfinity matrices.

Because we are dealing with infinity, we have to consider the kinds of things that we would consider when we have say an infinite series, but in the context of infinite dimensional vector spaces and matrices.

To do this we study things in terms of norms and we use a theory that is known as Hilbert-Space theory which studies convergence and gaining an understanding of infinite-dimensional spaces. If you want the kinds of things that cover this kind of thing, look at C*-Algebras, functional analysis and Hilbert-space theory.

Your question is not a simple one to answer because of the nature of infinity, but if I were to offer advice I would consider a useful place to start, I would suggest you look at how to define norms on these kinds of operators and then use the same kind of theory used in numerical linear algebra to see if the system itself is stable (I think the term is regular or something, but when I say stable I mean stable enough to obtain a solution).

If you need a place to start for this look for condition numbers and matrix norms or read a solid text on numerical linear algebra (should be in a decent linear algebra text).