Homogenous Non-Trivial Solution Theorem

If $n<k$, then every Homogenous Systems with $k$ variables and $n$ equations has a non-trivial solution.

Proof

1. Suppose $n<k$
2. Reduce the system of Echelon Form
3. There will be at most $n$ leading $1$s in the system.
4. Then, by defn of free variable, there will exists at least $(k-n)\ge 1$ free variables
5. Therefore, there will be infinitely many solns
6. Therefore, there will be a soln $\overrightarrow{x}\ne \overrightarrow{0}$