Algebra (02_XBALG)

• Coefficient : 3
• Hourly Volume: 62h (including 36h supervised)
  CTD : 36h supervised
  Out-of-schedule personal work : 26h

AATs Lists

Description

1. Reminders: Lines and planes in $\mathbb{R}^3$, vector product.
2. Matrices:
   • Matrix calculation
   • Invertible matrix
   • Change of basis
3. Determinant
4. Vector spaces :
   • Definition
   • Vector subspaces
   • Generating, free and related families. Bases and dimension of a vector space
   • Passage matrix
   • Sum of two sev
5. Linear applications
   • Kernel, image, injectivity, sujectivity
   • Matrix of a linear application in finite dimension
6. Reduction of endomorphisms :
   • Eigenvalues, eigenvectors, eigenspaces
   • Characteristic polynomial
   • Diagonalization
   • Applications: calculating the n-th power of a diagonalizable matrix, solving a linear differential system that can be written using a diagonalizable matrix.

Learning Outcomes AAv (AAv)

• AAv1 [heures: 18,B2,B3] : At the end of this course, the student knows how to implement the notions relating to vector spaces (vector subspaces, free family, generator, base) and is able to identify a vector space.

• AAv2 [heures: 15,B2,B3] (linear applications): At the end of this teaching, the student knows how to recognize or show that an application is linear, determine its kernel and its image. In finite dimension the student is able to explain the matrix of a linear application in a base.

• AAv3 [heures: 14,B1,B2,B3] (matrices): At the end of this course, the student masters matrix calculation (sum, product, inverse, determinant) and is able to recognize an algebra problem linear in a concrete situation.

• AAv4 [heures: 15,B1,B2,B3] (diagonalization): At the end of this course, the student knows how to diagonalize a square matrix and geometrically interpret the characteristic elements (eigenvalues, eigenvectors, eigensubspaces) . He can apply this to situations relating to the resolution of a system of recurring sequences, the calculation of the power of a matrix, the resolution of a linear differential system with constant coefficients.

Assessment methods

• a long test
• the average of several short tests

Key Words

Vector spaces, matrices, determinants, diagonalization.

Prerequisites

Concepts seen in mathematics specialty in high school

Resources

• Les mathématiques en licence : cours et exercices résolus. Tome 2 / Azoulay, Avignant, Auliac
• Algèbre linéaire : rappels de cours, questions de réflexion, exercices d'entraînement / Denmat, Héaulme
• Le succès en algèbre en fiches-méthodes / EL Kaabouchi
• Site Bibm@th (cours, exercices, quizz) https://www.bibmath.net
• Site Exo7 http://exo7.emath.fr/