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:

\(\ds \sum_{j \mathop = 1}^n \alpha_{i j} x_j\)

\(=\)

\(\ds \sum_{j \mathop = 1}^n \alpha_{i j} \times 0\)

\(\ds \)

\(=\)

\(\ds \sum_{j \mathop = 1}^n 0\)

\(\ds \)

\(=\)

\(\ds 0\)

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.

$\blacksquare$