# Definition: Elementary Gaussian Operations

Let an SLE be given in the form of its extended coefficient matrix. $$A|\beta:=\left(\begin{array}{ccc|c}\alpha_{11}& \ldots&\alpha_{1n}&\beta_1\\ \alpha_{21}& \ldots&\alpha_{2n}&\beta_2\\ \vdots&\vdots&\vdots&\vdots\\ \alpha_{m1}& \ldots&\alpha_{mn}&\beta_m\end{array}\right)$$

with elements $\beta_i,a_ij\in F$ of a field $F.$ The following are the elementary Gaussian operations:

### Row operations

1. Exchanging two rows of $A|\beta:$ $$R_j\leftrightarrow R_i.$$
2. Multiplying a row of $A|\beta$ by a number $c\in F$ with $c\neq 0$: $$c R_i\rightarrow R_i.$$
3. Adding a multiple of one row $R_i$ to another row $R_j:$ $$R_i+cR_j\rightarrow R_i.$$

### Column operation

1. Exchanging two columns of $A:$

$$C_j\leftrightarrow C_i.$$

Applications: 1
Definitions: 2
Lemmas: 3
Proofs: 4

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### References

#### Bibliography

1. Knabner, P; Barth, W.: "Lineare Algebra - Grundlagen und Anwendungen", Springer Spektrum, 2013