Why are all homogenous systems consistent? A linear system of form $A\vec{x}=\vec{0}$ is called homogeneous. Why are all homogenous systems consistent?

HINT $\ $ Zero is a root of every linear map $\rm\:A\:,\:$ since linear maps must  preserve $\rm\ 0 + 0 = 0\:,\ $ i.e.   
$$\rm\ A\ (0 + 0\ =\ 0)\ \ \to\ \ A(0) + A(0)\ =\ A(0)\ \ \Rightarrow\ \ A(0) = 0$$
More generally: monoid homomorphisms preserve idempotents, but the only idempotent element in a cancellative monoid is the neutral element $\rm\ a + a = a\ \Rightarrow\ a = 0\:.$
