Tight infinite matrices

We give a simple proof of a recent result of Gollin and Joó: 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 $a_{\phi (j),j}\mathrm{\ne }0$ for all $j\in J$.