Definition:Homogeneous Linear Equations

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:

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.