Homogeneous Linear Equations with More Unknowns than Equations

Theorem
Let $\alpha_{ij}$ be elements of a field $F$, where $1 \le i \le m, 1 \le j \le n$.

Let $n > m$.

Then there exist $x_1, x_2, \ldots, x_n \in F$ not all zero, such that:
 * $\displaystyle \forall i: 1 \le i \le m: \sum_{j=1}^n \alpha_{ij} x_j = 0$

Alternatively, this can be expressed as:

If $n > m$, the following system of homogeneous linear equations:

has at least one solution such that not all of $x_1, \ldots, x_n$ is zero.