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diff --git a/Master/texmf-dist/doc/latex/gauss/gauss-ex.tex b/Master/texmf-dist/doc/latex/gauss/gauss-ex.tex new file mode 100644 index 00000000000..35a6bafaf5b --- /dev/null +++ b/Master/texmf-dist/doc/latex/gauss/gauss-ex.tex @@ -0,0 +1,66 @@ +% +% This file is part of the gauss package by Manuel Kauers. +% Its purpose is to demonstrate the usage of the package. +% See the gauss-doc.dvi file for further information. +% +\documentclass{article} +\usepackage{amsmath} +\usepackage{gauss} +\usepackage{amssymb} % to import \leadsto + +\begin{document} + + \section{An Example for Gaussian Elimination} + + Consider the real matrix + \[ + A(t):=\begin{pmatrix} + 1 & 1 & 1 \\ + t & 2t & 2 \\ + t+1 & 0 & 2t + \end{pmatrix}, + \] + which depends on a real valued parameter~$t$. + We want to find all solutions of the homogenous linear system defined by $A$ + depending on the parameter~$t$. + + Therefore, we use the Algorithm Gauss: + \begin{alignat*}1 + A(t)=\begin{gmatrix}[p] + 1 & 1 & 1 \\ + t & 2t & 2 \\ + t+1 & 0 & 2t + \rowops + \add[-t]{0}{1} + \add[-(t+1)]{0}{2} + \end{gmatrix}\kern-6.23pt\leadsto& + \begin{gmatrix}[p] + 1 & 1 & 1 \\ + 0 & t & 2-t \\ + 0 & -t-1 & t-1 + \rowops + \add12 + \mult{2}{\cdot(-1)} + \end{gmatrix}\\ + \leadsto&\begin{gmatrix}[p] + 1 & 1 & 1 \\ + 0 & t & 2-t \\ + 0 & 1 & -1 + \rowops + \swap12 + \add[-t]12 + \mult2{:2} + \end{gmatrix}\\ + \leadsto&\begin{pmatrix} + 1 & 1 & 1 \\ + 0 & 1 & -1 \\ + 0 & 0 & 1 + \end{pmatrix}. + \end{alignat*} + + \subsection*{Conclusion} + + We have learned that the matrix $A$ defined above is regular for all real + valued~$t$, and we hopefully also have learned how to use the gauss package. + +\end{document}
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