Let \(F\) be a field and let
\[\begin{array}{ccl} \alpha_{11}x_1+\ldots+\alpha_{1n}x_n&=&\beta_1\\\ \alpha_{21}x_1+\ldots+\alpha_{2n}x_n&=&\beta_2\\ \vdots&\vdots&\vdots\\ \alpha_{m1}x_1+\ldots+\alpha_{mn}x_n&=&\beta_m\\ \end{array}\quad\quad( * )\]
be an SLE (system of linear equations) . The can write the coefficients \(\alpha_{ij}\in F\), \(i=1,\ldots,n\), \(j=1,\ldots,m\) as a matrix and call it the coefficient matrix $A$ of the SLE $( * ):$
$$A:=\pmatrix{\alpha_{11}& \ldots&\alpha_{1n}\\ \alpha_{21}& \ldots&\alpha_{2n}\\ \vdots&\vdots&\vdots\\ \alpha_{m1}& \ldots&\alpha_{mn}}$$
We also introduce another term we will need later. The tabular schema
$$A|\beta:= \left(\begin{array}{ccc|c}\alpha_{11}& \ldots&\alpha_{1n}&\beta_1\\ \alpha_{21}& \ldots&\alpha_{2n}&\beta_2\\ \vdots&\vdots&\vdots&\vdots\\ \alpha_{m1}& \ldots&\alpha_{mn}&\beta_m\end{array}\right)$$
is called the extended coefficient matrix of the SLE $( * ).$
Definitions: 1 2 3 4
Examples: 5 6 7 8
Lemmas: 9
Proofs: 10
Topics: 11
