What I want to show: A homogeneous system of linear equations, with fewer equations(m) than unknowns(n), has a non-trivial solution.
Lemma Used: Suppose you have a system of linear equations with n unknowns. Write the as $Ax=b$. Let $E$ be the systems reduced echelon matrix. If $E$ has k non-zero rows then then system will have $n-k>0$ free variables.
How do I prove this lemma (without using rank?)
Proof of what I want to show: So, if I have a system of equations $Ax=\textbf{0}$ with fewer equations then unknowns then I can reduce A to its unique Hermite matrix, $E$. (Hermite is its reduced echelon matrix). Further suppose $E$ is non-zero then Either it has a zero row or not. If it has a zero row then since the number of rows are less than the number of columns, we will have a solution with $n-k>0$ free variables, in which k is the number of non-zero rows (as $k\leq m<n$). If it does not have a zero row, then we will have $n-m>0$ solutions.
Is this proof correct? (Do not use rank, please)