Definition:Homogeneous Simultaneous Linear Equations

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Definition

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 if and only if:

$\forall i \in \set {1, 2, \ldots, m}: \beta_i = 0$


That is:

\(\ds \alpha_{1 1} x_1 + \alpha_{1 2} x_2 + \cdots + \alpha_{1 n} x_n\) \(=\) \(\ds 0\)
\(\ds \alpha_{2 1} x_1 + \alpha_{2 2} x_2 + \cdots + \alpha_{2 n} x_n\) \(=\) \(\ds 0\)
\(\ds \) \(\cdots\) \(\ds \)
\(\ds \alpha_{m 1} x_1 + \alpha_{m 2} x_2 + \cdots + \alpha_{m n} x_n\) \(=\) \(\ds 0\)


Also see

  • Results about simultaneous linear equations can be found here.


Sources