Volume 23, Issue 2, May 2016, Pages 455–465
David MAPENDANO BAHAGAZE1
1 Département de Mathématique-physique, Institut Supérieur Pédagogique d'Idjwi (ISP/IDJWI), Idjwi, Sud-kivu, RD Congo
Original language: French
Copyright © 2016 ISSR Journals. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The Rouché-Fontené theorem plays a fundamental role in the study of linear equations. It gives, in fact, the general rule for solving linear systems of the type n×p (that is to say, systems of linear equations in n unknowns p, (n,p)∈N^*×N^*). This rule is based on the determination and calculation of the respective values of critical characteristics or "trimmed" and, of course, on their list, given that, as they are all zero or one of them do not be the result of a linear system which they are extracted radically changes. The demonstration to prove that the set μ(p,q,r) real matrices p×q type and rank r is a variety diving is based on the same type of reasoning based on the identification and calculation tacking matrices determinants; It can therefore rightly be regarded as a unique illustration and / or application of an aspect of Rouché-Fontené theorem. The aim of this article is precisely to show how Rouché-Fontené theorem (mainly within linear algebra) is a scope in the dives varieties theory (rather under differential geometry).
Author Keywords: System of linear equations, matrix and determinant, determinant feature, bordered matrices , Compatibility, Rank, Immersion, Variety, Variety dives.
David MAPENDANO BAHAGAZE1
1 Département de Mathématique-physique, Institut Supérieur Pédagogique d'Idjwi (ISP/IDJWI), Idjwi, Sud-kivu, RD Congo
Original language: French
Copyright © 2016 ISSR Journals. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The Rouché-Fontené theorem plays a fundamental role in the study of linear equations. It gives, in fact, the general rule for solving linear systems of the type n×p (that is to say, systems of linear equations in n unknowns p, (n,p)∈N^*×N^*). This rule is based on the determination and calculation of the respective values of critical characteristics or "trimmed" and, of course, on their list, given that, as they are all zero or one of them do not be the result of a linear system which they are extracted radically changes. The demonstration to prove that the set μ(p,q,r) real matrices p×q type and rank r is a variety diving is based on the same type of reasoning based on the identification and calculation tacking matrices determinants; It can therefore rightly be regarded as a unique illustration and / or application of an aspect of Rouché-Fontené theorem. The aim of this article is precisely to show how Rouché-Fontené theorem (mainly within linear algebra) is a scope in the dives varieties theory (rather under differential geometry).
Author Keywords: System of linear equations, matrix and determinant, determinant feature, bordered matrices , Compatibility, Rank, Immersion, Variety, Variety dives.
Abstract: (french)
Le Théorème de Rouché-Fontené joue un rôle fondamental dans l’étude des d’équations linéaires. Il donne, en effet, la règle générale de résolution de systèmes linéaires de type n×p (c'est-à-dire des systemes de n équations linéaires à p inconnues,(n,p)∈N^*×N^*). Cette règle est fondée sur la détermination et le calcul des valeurs respectives de déterminants caractéristiques ou « bordés » et, bien entendu, sur leur énumération, étant donné le fait que, selon qu’ils soient tous nuls ou que l’un d’eux ne le soit pas, le résultat d’un système linéaire dont ils sont extraits change radicalement. La démonstration visant à prouver que l’ensemble μ(p,q,r) des matrices réelles de type p×q et de rang r est une variété plongée s’appuie sur ce même type de raisonnement basé sur l’identification et le calcul des déterminants de matrices bordées ; Elle peut donc, à juste titre, être considérée comme une singulière illustration et /ou application d’un aspect du théorème de Rouché-Fontené. L’enjeu du présent article est précisément de montrer comment le Théorème de Rouché-Fontené (relevant principalement de l’algèbre linéaire) trouve un champ d’application dans la théorie de variétés plongées (relevant plutôt de la géométrie différentielle).
Author Keywords: Système d’équations linéaires, Matrice et Déterminant, Déterminant caractéristique, matrices bordées, Compatibilité, Rang, Immersion, Variétés, Variétés plongées.
How to Cite this Article
David MAPENDANO BAHAGAZE, “SUR UNE APPLICATION « SINGULIÈRE » DU THÉORÈME DE ROUCHÉ-FONTENÉ EN GÉOMÉTRIE DIFFÉRENTIELLE,” International Journal of Innovation and Scientific Research, vol. 23, no. 2, pp. 455–465, May 2016.
