A homogeneous system with more variables than equations has infinitely many solutions.

Theorem: A homogeneous system $Ax = 0$ with more variables than equations ($n > m$) always has infinitely many solutions.

This follows from two facts:

1. Homogeneous systems are always consistent (the trivial solution $x = 0$ always exists)
2. A consistent system with $n > m$ has at least one free variable

Combined: a homogeneous system with $n > m$ has a free variable, which means it has infinitely many solutions.

Example:
$$\begin{cases} x_1 + x_2 + x_3 = 0 \\ x_1 - x_2 + 2x_3 = 0 \end{cases}$$

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

This theorem is frequently used to prove that sets of vectors are linearly dependent.