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

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  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