Tight infinite matrices

TL;DR

This paper provides a straightforward proof of a recent result concerning infinite matrices, showing that if a system of linear equations has only the trivial solution, then a specific non-zero pattern exists in the matrix.

Contribution

The paper introduces a simple proof of a known theorem about the structure of infinite matrices with trivial solution spaces.

Findings

Existence of an injection mapping columns to rows with non-zero entries

Simplification of the proof of a recent theorem

Clarification of the structure of matrices with trivial solutions

Abstract

We give a simple proof of a recent result of Gollin and Jo\'o: if a possibly infinite system of homogeneous linear equations $A\vec{x} = \vec{0}$, where $A = (a_{i, j})$ is an $I \times J$ matrix, has only the trivial solution, then there exists an injection $\phi: J \to I$, such that $a_{\phi(j), j} \neq 0$ for all $j \in J$.