Matching variables to equations in infinite linear equation systems

TL;DR

This paper generalizes a fundamental linear algebra result to infinite systems, showing that if such a system has only the trivial solution, variables can be injectively matched to equations they appear in.

Contribution

It proves a new theorem extending finite linear algebra principles to infinite systems with finitely many variables per equation.

Findings

Infinite homogeneous systems with only trivial solutions allow an injection from variables to equations.

The result generalizes the finite case to certain infinite systems.

Provides a structural insight into the variable-equation relationships in infinite systems.

Abstract

A fundamental result in linear algebra states that if a homogenous linear equation system has only the trivial solution, then there are at most as many variables as equations. We prove the following generalisation of this phenomenon. If a possibly infinite homogenous linear equation system with finitely many variables in each equation has only the trivial solution, then there exists an injection from the variables to the equations that maps each variable to an equation in which it appears.