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:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 0\)	\(=\)	\(\displaystyle \alpha_{11} x_1 + \alpha_{12} x_2 + \cdots + \alpha_{1n} x_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 0\)	\(=\)	\(\displaystyle \alpha_{21} x_1 + \alpha_{22} x_2 + \cdots + \alpha_{2n} x_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\cdots\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 0\)	\(=\)	\(\displaystyle \alpha_{m1} x_1 + \alpha_{m2} x_2 + \cdots + \alpha_{mn} x_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

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

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): Chapter $1 \ \S 4$