homogeneous system

$A\mathbf x = \mathbf 0$ — a system whose right-hand side is entirely zero.

A system $A\mathbf x = \mathbf 0$ , every constant term zero, is homogeneous. It is never inconsistent, since $\mathbf x = \mathbf 0$ (the trivial solution) always works. If every column of $A$ has a pivot, that trivial solution is unique; if some column is free, infinitely many nontrivial solutions exist, and the solution set is always a span — closed under addition and scalar multiplication.