\begin{definition}[Definition:Homogeneous Linear Equations]
A '''system of homogeneous linear equations''' is a set of simultaneous linear equations:
:$\ds \forall i \in \closedint 1 m: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
such that all the $\beta_i$ are equal to zero:
:$\ds \forall i \in \closedint 1 m : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$
That is:
{{begin-eqn}}
{{eqn | l = 0
      | r = \alpha_{11} x_1 + \alpha_{12} x_2 + \cdots + \alpha_{1n} x_n
}}
{{eqn | l = 0
      | r = \alpha_{21} x_1 + \alpha_{22} x_2 + \cdots + \alpha_{2n} x_n
}}
{{eqn | o = \cdots}}
{{eqn | l = 0
      | r = \alpha_{m1} x_1 + \alpha_{m2} x_2 + \cdots + \alpha_{mn} x_n
}}
{{end-eqn}}
\end{definition}