In Hoffman and Kunze they have two exercises where they ask to show that if two homogeneous linear systems have the exact same solutions then they have the same row-reduced echelon form.
They first ask to prove it in the case of $2\times 2$ (Exercise 1.2.6) and then they ask to prove it in the case $2\times 3$ (Exercise 1.4.10). I am able to prove it in both of these special cases, but as far as I can tell Hoffman and Kunze never tell us whether or not this is true in general.
So that's my question, is this true in general? And if not, can anybody provide a counter-example? Thank you!