We know that we can apply [[EROs]] to any augmented matrix into [[REF]]. 

Suppose the system has $n$ equations and $m$ variables, and let $k$ be the number of non-zero rows in REF. Also suppose the system is consistent: then the REF has no row of the form $[0~0~0~\dots~1]$.

{{ ::ge-obs.png?nolink&800 |}}

  * $k\le n$, because there are only $n$ rows in the whole matrix
  * $k$ is precisely the number of [[leading variables]]. So $k$ is no bigger $m$, the total number of variables; in symbols, we have $k\le m$.
  * All the other variables are free variables, so $$ \text{$m-k$ is the number of free variables.} $$

What does this tell us about the set of solutions? For example, how many solutions are there?

=== Observation 1: free variables and the number of solutions ===

For consistent systems, this shows that:

  * //either// $k=m$;
    * so $m-k=0$
    * there are no free variables
    * the system has one solution and no more
    * We say it has a **unique solution**.
  * //or// $k<m$
    * so $m-k>0$
    * there is at least one free variable
    * so the system has **infinitely many solutions** (one for each value of each free variable)
    * The number of free variables, $m-k$, is called the [[wp>dimension]] of the solution set.

=== Observation 2: systems with fewer equations than variables ===

For consistent systems where $n<m$ (fewer equations than variables):

  * $k\le n < m$, so $k<m$. 
  * So there is at least one free variable.
  * So in this situation we //always// have infinitely many solutions.

/*
=== Observation 3: homogeneous systems ===

A linear system of the following form is called **homogeneous**:

\begin{align*} a_{11}x_1+a_{12}x_2+\dots+a_{1m}x_m&=0\\
a_{21}x_1+a_{22}x_2+\dots+a_{2m}x_m&=0\\
\hphantom{a_{11}}\vdots \hphantom{x_1+a_{22}}\vdots\hphantom{x_2+\dots+{}a_{nn}} \vdots\  & \hphantom{{}={}\!} \vdots\\
a_{n1}x_1+a_{n2}x_2+\dots+a_{nm}x_m&=0
\end{align*}

The important thing is that all the right hand sides are $0$. 

  * This system is necessarily consistent, since $(0,0,0,\dots,0)$ is a solution.
  * If $n<m$, then by the previous observation there are infinitely many solutions.
*/