A homogeneous system has a nontrivial solution if and only if it has a free variable.

Theorem: A homogeneous system $Ax = 0$ has a nontrivial solution if and only if it has at least one free variable.

Intuition: In the echelon form of $A$, if every variable is a basic variable (has a pivot), then the only solution is $x = 0$ (trivial). If at least one variable is free, it can be assigned a nonzero value, which propagates through the system to produce a nonzero solution.

Corollary: A homogeneous system with more variables than equations ($n > m$) always has a nontrivial solution, because there can be at most $m$ pivots, leaving at least $n - m > 0$ free variables.

Example:
$$\begin{cases} x + y + z = 0 \\ 2x + 3y - z = 0 \end{cases}$$

2 equations, 3 variables → at least 1 free variable → infinitely many nontrivial solutions.