Definition:Homogeneous Linear Equations
Jump to navigation
Jump to search
This page is about Homogeneous Linear Equations. For other uses, see Homogeneous.
Definition
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:
\(\ds 0\) | \(=\) | \(\ds \alpha_{11} x_1 + \alpha_{12} x_2 + \cdots + \alpha_{1n} x_n\) | ||||||||||||
\(\ds 0\) | \(=\) | \(\ds \alpha_{21} x_1 + \alpha_{22} x_2 + \cdots + \alpha_{2n} x_n\) | ||||||||||||
\(\ds \) | \(\cdots\) | \(\ds \) | ||||||||||||
\(\ds 0\) | \(=\) | \(\ds \alpha_{m1} x_1 + \alpha_{m2} x_2 + \cdots + \alpha_{mn} x_n\) |
Matrix Representation
A system of homogeneous linear equations 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
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 4$. Vector Spaces
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): homogeneous set of linear equations