Homogeneous system of linear equations

Definition

Let $F$ be a field. A system of linear equations over $F$ is homogeneous when each of its constant terms is $0$. In matrix form, a homogeneous system is an equation $Ax = 0$, where $A$ is an $m \times n$ matrix over $F$ and $x = (x_1, \dots, x_n)^T$.

Remarks

The tuple $(0, \dots, 0)$ is always a solution, the trivial solution. The solution set of $Ax = 0$ is a linear subspace of $F^n$, the null space of $A$. If a system $Ax = b$ has a solution $p$, then its solution set is $\{p + v : Av = 0\}$. A homogeneous system with more unknowns than equations has a nontrivial solution.

Explanations

L1 human

A homogeneous system has zero on the right side of every equation. Setting every unknown to zero always works, so the question is whether other solutions exist. The sum of two solutions is again a solution, and so is any multiple of a solution.