quartz/content/vault/homogeneous.md

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#math/linear_algebra

definition

A system of linear equations is said to be ==homogeneous== if it can be written as an augmented matrix of the form [A \mid \mathbf{0}], where A is an m \times n matrix and \mathbf{0} is the zero vector in \mathbb{R}^{m}. ^1652734750442

Example: The following system of equations is homogeneous (all of the constants on the right-hand side are zero):

\begin{array}{r}
3 x_{1}+5 x_{2}-4 x_{3}=0 \\
-3 x_{1}-2 x_{2}+4 x_{3}=0 \\
6 x_{1}+x_{2}-8 x_{3}=0
\end{array}

An example matrix: $\left[\begin{array}{ccc|c} 3 & 5 & -4 & 0 \ -3 & -2 & 4 & 0 \ 6 & 1 & -8 & 0 \end{array}\right]$