\begin{definition}[Definition:Homogeneous Simultaneous Linear Equations]
A '''system of simultaneous linear equations''':
:$\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
is referred to as '''homogeneous''' {{iff}}:
:$\forall i \in \set {1, 2, \ldots, m}: \beta_i = 0$
That is:
{{begin-eqn}}
{{eqn | r = 0
      | l = \alpha_{1 1} x_1 + \alpha_{1 2} x_2 + \cdots + \alpha_{1 n} x_n
}}
{{eqn | r = 0
      | l = \alpha_{2 1} x_1 + \alpha_{2 2} x_2 + \cdots + \alpha_{2 n} x_n
}}
{{eqn | o = \cdots}}
{{eqn | r = 0
      | l = \alpha_{m 1} x_1 + \alpha_{m 2} x_2 + \cdots + \alpha_{m n} x_n
}}
{{end-eqn}}
\end{definition}