How Can I Infer the Expression of A in a Non-Trivial Solution?

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Discussion Overview

The discussion revolves around solving a system of linear equations to find a non-trivial solution. Participants explore the conditions under which a non-trivial solution exists and the implications for expressing one variable in terms of others within a 4x4 system of equations.

Discussion Character

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

Main Points Raised

  • One participant states that for a non-trivial solution to exist, the determinant must be zero, leading to the equation (f(x))*A=0.
  • Another participant questions the definitions of f(x) and A, suggesting they might be matrices or polynomials.
  • A participant clarifies that they are dealing with 4 equations and 4 unknowns, ultimately reducing it to an equation of the form (2g+h)*(x)=0.
  • It is proposed that if (2g+h)=0, then any value of x would satisfy the equation, raising the question of whether an expression for x in terms of g and h can be inferred.
  • One participant suggests that adding a constraint might help avoid an infinite solution case, while expressing reluctance to change the original constraints.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the variables and the implications of the conditions for a non-trivial solution. There is no consensus on whether an expression for x can be definitively inferred from the given conditions.

Contextual Notes

Participants acknowledge the need for clearer definitions and the potential for ambiguity in the mathematical framework being discussed. The discussion reflects varying levels of mathematical background among participants.

Who May Find This Useful

This discussion may be of interest to individuals exploring linear algebra, particularly in the context of solving systems of equations and understanding the conditions for non-trivial solutions.

Ribena
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Hi all,

So I've been trying to solve a system of 'linear' equations and I understand that for a non trivial solution to exist you have to get the determinant to reduce to zero.

Given that you have an equation which takes the form

(f(x))*A=0, where f(x) is an arbitary function and A is a constant, the solution is either f(x)=0 or A=0.

Now say that for a non trivial solution, I'd require that f(x)=0, is there any methods in general in which I can obtain an expression for A? Let's assume it's a 4 by 4 system of equation.
 
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What are f(x) and A? Matrices?
 
Erland said:
What are f(x) and A? Matrices?

Polynomials.
 
Ribena said:
Polynomials.
How can it then be a 4x4 system of equations?

No, you must more clearly specify what you mean...
 
Erland said:
How can it then be a 4x4 system of equations?

No, you must more clearly specify what you mean...

Ah, I get what you mean now. I'm not from a mathematical background so do bear with me is some of my definitions are not precise. So I basically have 4 equations with 4 unknowns to solve let's assume that they are w,x,y,z. By eliminating the variables, I've managed to bring them down a single equation that looks something like (2g+h)*(x)=0 where g and h are predefined constants.

So for a non trivial solution to exist, then (2g+h)=0. Will it be possible then to know the expression for x in terms of g and h (if one exists at all)?

Hopefully it's not too confusing
 
Ribena said:
So for a non trivial solution to exist, then (2g+h)=0. Will it be possible then to know the expression for x in terms of g and h (if one exists at all)?
But if 2g+h=0, then any value of x will do.
 
Erland said:
But if 2g+h=0, then any value of x will do.

Exactly, so I was wondering if there was any way in general to infer the expression of x. I think the only way is to perhaps add a constraint that does not result in an infinite solution case, but if possible, I didn't want to change the original constraints.

Just wanted to seek some opinions to see if there actually is some method that I was unaware of.
 

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