Tight infinite matrices

We give a simple proof of a recent result of Gollin and Joo (Matching variables to equations in infinite linear equation systems. Linear Algebra Appl. 2023;660:40-46. doi: 10.1016/j.laa.2022.12.002): if a possibly infinite system of homogeneous linear equations $ A\vec {x} = \vec {0} $ Ax ->=0 ->, where $ A = (a_{i, j}) $ A=(ai,j) is an $ I \times J $ IxJ matrix, has only the trivial solution, then there exists an injection $ \phi : J \to I $ phi:J -> I, such that $ a_{\phi (j), j} \neq 0 $ a phi(j),j not equal 0 for all $ j \in J $ j is an element of J.