Homogenous Systems

The system $Ax=0$ is called homogenous. Homogenous systems are always consistent: $x=0$ is always a solution. If $Ax=0$ has a non-trivial solution $x\ne 0$, then it has infinitely many solutions.

Corollary:

1. $Ax=0$ has a nontrivial solution $x\ne 0$ if and only if $\mathrm{rank}A<n$.
2. If $m<n⟹Ax=0$ always has a nontrivial solution
3. If $m=n⟹Ax=0$ has nontrivial solution if and only if $A$ is singular.