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summary Matrices and Linear Transformations

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Course
Algebre (ALGBR.1)

Institution
Université De Montréal ( )

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...

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Uploaded on March 10, 2023
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Cours d’algèbre 1
Filière MIP-MIPC
premier semestre

........................................
Chapitre : Calcul matriciel et systèmes linéa
((Chapitre complet))
[Séance ..]

Karim KREIT

FST-UCA

,Copyright © 2020-2021 Karim KREIT

Copying

All rights reserved.

Art. No xxxxx
ISBN xxx–xx–xxxx–xx–x
Edition 0.0

Published by FST-UCA

,Table des matières

1 Matrices (Définitions et propriétés). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1 Calcul Matriciel ....................................................................................... 4
1.1.1 Définition et notations ...................................................................... 4
1.1.2 Matrices particulières ....................................................................... 5
1.1.3 Opérations sur les matrices .............................................................. 8
1.2 Quelques propriétés des matrices carrées ................................................ 14
1.2.1 Puissance d’une matrice ................................................................. 14
1.2.2 Matrices symétriques-Matrices antisymétriques ................................ 16
1.2.3 La trace ........................................................................................ 17
1.2.4 Matrices inversibles ....................................................................... 19

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

a1,1 a1,2 ... a1,j ... a1,p 
 
a2,1 a2,2 ... a2,j ... a2,p 
 
 
 . . . ... ... ... ... . . . 
A = 
 ai,1 ai,2 ... ai,j ... ai,p 

 
 . . . ... ... ... ... . . . 

an,1 an,2 ... an,j ... an,p


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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