# Definition: Systems of Linear Equations with many Unknowns

Let $$F$$ be a field. A system of linear equations (SLE) with $$n$$ unknowns $$x_1,\ldots,x_n$$ and ($$m\ge 1$$) linear equations is given by

$\begin{array}{ccl} \alpha_{11}x_1+\ldots+\alpha_{1n}x_n&=&\beta_1\\\ \alpha_{21}x_1+\ldots+\alpha_{2n}x_n&=&\beta_2\\ \vdots&\vdots&\vdots\\ \alpha_{m1}x_1+\ldots+\alpha_{mn}x_n&=&\beta_m\\ \end{array}\quad\quad( * )$

with $$\alpha_{ij}\in F$$, $$i=1,\ldots,n$$, $$j=1,\ldots,m$$ and some $$\beta_1,\ldots,\beta_m\in F$$.

The SLE $( * )$ is called * homogeneous if $\beta_1=0,\ldots,\beta_n=0,$ * inhomogeneous if $\beta_i\neq 0$ for at least one $i\in\{1,\ldots,m\}.$

The SLE $( * )$ has a solution, if there exist numbers $$a_1,\ldots,a_n\in F$$ such that for the replacement $$x_1:=a_1,\ldots,x_n:=a_n$$ all linear equations in the SLE are simultaneously fulfilled.

The solution of a homogeneous SLE is called trivial if $$a_i=0$$ for all $$i=1,\ldots,n$$, otherwise (if a solution with at least one $$\beta_i\neq 0$$ exists), it is called non-trivial.

### Indexing Convention

In the double-index $$ij$$ of the coefficients $$\alpha_{ij}$$, by convention, the first index $$i$$ denotes the row, the second index $$j$$ denotes the column of the system.

Applications: 1
Chapters: 2
Definitions: 3 4 5
Examples: 6
Lemmas: 7
Proofs: 8 9 10
Sections: 11
Theorems: 12
Topics: 13

Thank you to the contributors under CC BY-SA 4.0!

Github:

### References

#### Bibliography

1. Koecher Max: "Lineare Algebra und analytische Geometrie", Springer-Verlag Berlin Heidelberg New York, 1992, 3rd Volume
2. Knabner, P; Barth, W.: "Lineare Algebra - Grundlagen und Anwendungen", Springer Spektrum, 2013