(related to Section: Solving Simple Systems of Linear Equations)
Our first example is a rather degenerated SLE with an extended coefficient matrix consisting of a zero matrix:
$$\left(\begin{array}{ccc|c}0& \ldots&0&\beta_1\\ 0& \ldots&0&\beta_2\\ \vdots&\vdots&\vdots&\vdots\\ 0& \ldots&0&\beta_m\end{array}\right)$$
Each linear equation in such an SLE has the form
$$0x_1+0x_2+\ldots+0x_n=\beta_j$$
for $j=1,\ldots,m$ and we have seen in a corollary that these equations have * either no solutions at all if $\beta_j\neq 0$ * or infinitely many solutions with arbitrary choices for the unknowns $x_1,\ldots,x_n,$ if $\beta_j=0.$
From this, we can conclude that the above SLE will have * either no solutions at all, if $\beta_j\neq 0$ for at least one $j$ with $1\le j \le m,$ * or infinitely many solutions with arbitrary choices for the unknowns $x_1,\ldots,x_n,$ if $\beta_j=0$ for all $j$ with $1\le j \le m.$
Definitions: 1