A course on matrices and vector spaces, also known as linear algebra, is a fundamental subject in mathematics that deals with linear equations, systems of equations, and the properties of vectors and matrices.
The course starts with an introduction to matrices, which are rectangular arrays of nu...
2 Résolution des systèmes linéaires (Pivot de Gauss) . . . . . . . . . . . . . . . . . . . 23
2.0.1 Les matrices élémentaires .............................................................. 23
2.1 Matrices échelonnées ............................................................................. 25
2.1.1 Équivalence à une matrice échelonnée ............................................. 26
2.1.2 Inverse d’une matrice - Méthode Gausse .......................................... 29
2.2 Systèmes linéaires ................................................................................. 32
2.2.1 Matrices et systèmes linéaires ........................................................ 32
2.2.2 Matrices inversibles et systèmes linéaires ........................................ 32
3
, 1. Matrices (Définitions et propriétés).
1.1 Calcul Matriciel ........................................................ 4
1.2 Quelques propriétés des matrices carrées .................... 14
Dans ce chapitre, K désigne un corps. On peut penser à Q, R ou C.
1.1 Calcul Matriciel
1.1.1 Définition et notations
Definition 1.1.
Soit (n, p) ∈ N∗ .
On appelle matrice de type (n, p) ou de format n×p à coefficients dans K, un tableau
rectangulaire A à n lignes et p colonnes d’éléments de K, c’est-à-dire
Et en abrégé: A = ai,j 16i6n ou plus simplement (aij ) lorsqu’il n’y a pas de confu-
16j6p
sion.
• Les ai,j sont appelés coefficients de la matrice A. L’élément ai,j correspond a
la valeur de l’intersection de la ligne i et de la colonne j.
• On dit que A est de taille n × p.
L’ensemble des matrices à n lignes et p colonnes à coefficients dans K est noté
Mn,p (K). Si K = R les éléments de Mn,p (R) sont appelés matrices réelles.
Exemple 1.1 !
1 −2 5
A=
0 3 7
est une matrice 2 × 3 avec, par exemple, a1,1 = 1 et a2,3 = 7.
4
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