I'm trying to prove that if the systems $AX = 0$ and $BX=0$ are equivalent, then the matrices $A$ and $B$ are row equivalent. Proving the converse was very simple, but this one seems harder. I saw a few answers to similar questions posted here before, but they were done using some higher level concepts which I am unaware of right now (such as orthogonal subspaces etc.). I am familiar with the concepts of linear equations, ranks of matrices, and the basic idea of subspaces(only, definition, dimension, basis etc.)

Any help would be appreciated.........