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 $a_{\phi(j), j} \neq 0$ for all $j \in J$.