Homogeneous Linear System

A homogeneous linear system is a linear system in which all of the constant terms are zero. In other words, it is when the associated matrix equation has $\vec{b}$ is the zero vector.

$$A\vec{x}=\vec{0}$$

These equations always have the zero (trivial) solution (where $\vec{x}$ is a zero vector)

If there are free variables, then there are more solutions in addition to the trivial solution. Otherwise, only the trivial solution exists.

Note: if there is one nontrivial solution, then there must be a family of solutions (ref: null space of a matrix)