Theorem. Every homogeneous system has either exactly one solution or infinitely many solutions. If a homogeneous system has more unknowns than equations, then it has infinitely many solutions.

Proof. Indeed, as we said the first statement directly follows from the theorem about solutions of systems of linear equations. The second statement follows from the fact that if the number of unknowns is bigger than the number of equations then some unknowns are free, and as we saw, if we have a free unknown, we get infinitely many solutions.