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

**Theorem (over an infinite field, such as the reals or complexes):** 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.

Over a finite field with $q$ elements, the same free-variable argument gives $q^{n-r}$ solutions rather than infinitely many.Copy