Definition:Homogeneous Linear Equations

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Definition

A system of homogeneous linear equations is a set of simultaneous linear equations:

$\displaystyle \forall i \in \left[{1 \,.\,.\, m}\right] : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$

such that all the $\beta_i$ are equal to zero:

$\displaystyle \forall i \in \left[{1 \,.\,.\, m}\right] : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$

That is:

\(\displaystyle 0\) \(=\) \(\displaystyle \alpha_{11} x_1 + \alpha_{12} x_2 + \cdots + \alpha_{1n} x_n\) $\quad$ $\quad$
\(\displaystyle 0\) \(=\) \(\displaystyle \alpha_{21} x_1 + \alpha_{22} x_2 + \cdots + \alpha_{2n} x_n\) $\quad$ $\quad$
\(\displaystyle \) \(\cdots\) \(\displaystyle \) $\quad$ $\quad$
\(\displaystyle 0\) \(=\) \(\displaystyle \alpha_{m1} x_1 + \alpha_{m2} x_2 + \cdots + \alpha_{mn} x_n\) $\quad$ $\quad$


Matrix Representation

Such a system is often expressed as:

$ \mathbf A \mathbf x = \mathbf 0$

where:

$ \mathbf A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix}$, $\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$, $\mathbf 0 = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$

are matrices.


Sources