Trivial Solution to System of Homogeneous Simultaneous Linear Equations is Solution

Theorem
Let $S$ be a system of homogeneous simultaneous linear equations:


 * $\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$

Consider the trivial solution to $A$:
 * $\tuple {x_1, x_2, \ldots, x_n}$

such that:
 * $\forall j \in \set {1, 2, \ldots, n}: x_j = 0$

Then the trivial solution is indeed a solution to $S$.

Proof
Let $i \in \set {1, 2, \ldots, m}$.

We have:

This holds for all $i \in \set {1, 2, \ldots, m}$.

Hence:
 * $\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$

and the result follows.