% This is lecture no. 7 of MH1200/: Linear Algebra I.

\chapter{Homogeneous Systems}%
\label{cha:homogeneous_systems}

A system of linear equations is said to be homogeneous is the constant term in each equation is zero:
\begin{align*}
	x_1 + x_2 + x_3 & = 0\\
	-x_1 + 2x_2 -3x_3 & = 0\\
	2x_1 + 4x_2 + 6x_3 & = 0
\end{align*}

A homogeneous system of linear equations is always solved by setting every variable to 0.
In other words: the $(0,\ldots,0)\in \mathbb{R}^n$ is always a solution to the system if the system has $n$ variables.
This solution is called the trivial solution.
Any other solution is called a non-trivial solution.

\begin{mdframed}
	\begin{proof}
		A homogeneous linear system of $m$ equations in $n$ variables $x_1,\ldots,x_n$ is of the form:
		\begin{align*}
			a_{11}x_1 + a_{12}x_2 + \ldots + a_{1n}x_n & = 0\\
								   & \;\; \vdots\\
			a_{m1}x_1 + a_{m2}x_2 + \ldots + a_{mn}x_n & = 0
		\end{align*}
		If we set every variable to 0, then every expression on the LHS evaluates to 0, and so every equation in the system is satisfied, simultaneously.
	\end{proof}
\end{mdframed}

If we think of the set of solutions as some subset of $\mathbb{R}^n$ then the set of solutions must include the origin.

\section{Counting Solutions}%
\label{sec:counting_solutions}

Because a homogeneous system always contains the trivial solutions, it is always consistent.
The usual three possibilities becomes only two possibilities:
\begin{enumerate}
	\item The system only has the trivial solution.
	\item The system has infinitely many solutions including the trivial solution.
\end{enumerate}

There are three cases for homogeneous systems regarding the relation of the number of equations and the number of variables:
\begin{enumerate}
	\item The case (\# equations) $<$ (\# variables).
	\item The case (\# equations) $=$ (\# variables).
	\item The case (\# equations) $>$ (\# variables).
\end{enumerate}

\noindent
Case 1: A homogeneous system with (\# equations) $<$ (\# variables).
This case will always have an infinite number of solutions, including the trivial solution.
The result of Gaussian elimination will result in some column without a leading entry, which implies a free parameter.

\bigbreak
\noindent
Case 2: A homogeneous system with (\# equations) $=$ (\# variables).
This case can either have an infinite number of solutions including the trivial solution, or have only the trivial solution.

\bigbreak
\noindent
Case 3: A homogeneous system with (\# equations) $>$ (\# variables).
This case can either have an infinite number of solutions including the trivial solution, or have only the trivial solution.
