> [!summary] A homogeneous linear equation is a linear equation that equals zero. > **Key Equations & Solutions:** > System of homogeneous linear equations: $A \vec{x}=0$ > Solutions: $\text{rank}(A) = n$ one solution > $\text{rank(A)} < n$ infinite solutions >[!info]+ Read Time **⏱ 2 mins** # Definition A homogeneous linear equation is a [[Linear Equations|linear equation]] where the solution value is equal to 0. For example, $a_{1}x_{1}+a_{2}x_{2}\dots a_{n}x_{n}=0$ is a homogeneous linear equation. In a [[Matrix Notation|matrix]], a [[Sets|set]] of linear equations is called a homogeneous system with $m$ equations and $n$ unknowns $ \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} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} $ This matrix can also be simplified, letting the coefficient matrix be $A$ and the [[Zero Vector|zero vector]] as $\vec{0}$ or $ A\vec{x}=\vec{0} $ # Solutions The number of possible solutions in a homogeneous linear equation is dependent on the number of variables ($n$) and the [[Rank|rank]] of the matrix ($A$). If the $\text{rank}(A) = n$, this means that every variable has a [[Leading 1s|pivot]] and the only solution is the **trivial solution** $ \vec{x}=\vec{0} $ If the $\text{rank(A)} < n$, there is at least one free variable that you can set to any real number. This means the result is **non-trivial** and there are **infinitely many solutions** The solution set of a homogeneous system of $m$ equations and $n$ variables is a [[Subspace|subspace]] of $\mathbb{R}^n$ and is called the **solution space** of a system. > [!note]- Why does $\text{Rank}(A)<n$ mean infinite solutions Take the example $x+2y+3z=0$ and suppose a vector equation is wanted for the solved system. So the matrix of this is > > $ \begin{bmatrix} 1 & 2 & 3 > \end{bmatrix} > \begin{bmatrix} x \\ y \\ z > \end{bmatrix} = > \begin{bmatrix} 0 > \end{bmatrix} > $ > > And the rank of the matrix is 1, where there are 3 variables. So the rank is less than the number of variables. If the question was to make [[Span|a vector equation]] of this equation [[Solving a Linear System Using Parametrization|solving this using parametrization]] must be done > This leads to the following outcome > $ > \begin{array}{c} x = -2s -3t \\ \\ \text{So the vector equation is } \\ \vec{x} = s \\ > \begin{bmatrix} -2 \\ 1 \\ 0 > \end{bmatrix} + t \begin{bmatrix} -3 \\ 0 \\ 1 \end{bmatrix}, \space t,s \in \mathbb{R} \end{array} > $ Since $s$ and $t$ can be any [[Real Numbers|real number]] there are **infinite vectors** that can be made