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Observations about Gaussian elimination
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 $[0~0~0~\dots~1]$.
- $k\le n$, 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 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.