Presentation
This course consists on the introduction of the basis of linear algebra
Objectives
The goals for this course are using matrices and also understanding them. Here are key computations and some of the ideas behind them:
Solving Ax = b for square systems by elimination (pivots, multipliers, back substitution, invertibility of A)
Complete solution to Ax = b (column space containing b, rank of A, nullspace of A and special solutions to Ax = 0 from row reduced R)
Basis and dimension (bases for the four fundamental subspaces)
Properties of determinants (leading to the cofactor formula and the sum over all n! permutations, applications to inv(A) and volume)
Eigenvalues and eigenvectors (diagonalizing A, computing powers A^k)
Duration:
9h
Content
1. Algebra basics
The main goal of this part is to learn (or revise) the basics of linear Algebra: matrices, determinants, linear applications, eigenvectors and eigenvalues, characteristic polynomials, diagonalization of matrices, reduction to triangular form, theorem of Cayley-Hamilton. A secondary but also important goal of this course is to learn everyday French mathematical terminology (notation, terminology, expressions) so to be able to read mathematical texts written in French and understand lectures in French. To achieve both goals equally well, we have organized this course so as to be a thorough study of the complete chapter 18 of the French math book "Algèbre : premier cycle et préparation aux Grandes Écoles", by Michel Queysanne, Librairie Armand Colin, 1964. The most important exercices in this chapter will be covered.
Organization
Examination
There will a final exam of 1h30. The use of calculators or notes is not permitted during the exams.
Scheduled activities
- C1 - AB (2h) The geometry of linear equations
- PC1 - AB (1h) The geometry of linear equations
- C2 - AB (2h) Matrix operations and inverses
- PC2 - AB (1h) Matrix operations and inverses
- C3 - AB (2h) Eigenvalues and eigenvectors
- PC3 - AB (1h) Eigenvalues and eigenvectors
Team
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