Homogeneous System

For a matrix represented as $\left[\begin{array}{c}A|b\end{array}\right]$. Where $A$ is the coefficient matrix and $b$ is the vector of solutions, If $\vec{b}$ is the $\vec{0}$, this is a homogenous matrix representing a homogeneous system.

Example

Homogeneous Form

You can represent a 2D point in homogeneous form (homogeneous coordinates) which represents that 2D point in a higher order. The point of representing points in this form, is so that Geometrical Transformations can be applied using matrix multiplication.

For example, $(x,y)\to (x,y,w)$ where $w$ is the homogeneous coordinate and is basically like a scaling factor. So $(x,y)⟹(x,y,1)$ and the cartesian coordinates can be recovered by $(x,y)=\left(\frac{x}{w},\frac{y}{w}\right)$.