Mathematical Tools 3

This course unit is intended for first-year PCSI students. It provides an introduction to linear algebra and the solution of linear differential systems (matrix calculus, solution of linear systems, eigenvalues and diagonalization, solution of linear differential systems).

- Matrices: operations, power, properties, calculation. Notable matrices. The vector space of matrices. The concept of vector subspace. The space^Rn viewed as the space of column matrices.

- Linear systems and pivot method. Solution space.

- Determinants in dimensions 2 and 3 - Invertible matrices - Application to linear systems.

-^Rn bases - coordinates. Intuitive notion of dimension. Change of basis - transition matrix.

- Linear applications^Rp →^Rn. Matrix relative to a basis. Basis change formula.

- Elementary diagonalization (exercises and examples will be limited to 2×2 and 3×3 matrices).

- Application of diagonalization to the solution of linear differential systems.

- (Optional: Matrix exponential. Introduction to nonlinear differential systems.)

Prerequisites*: High school mathematics program + EU Mathematical Tools 1.

Recommended prerequisites*: Advanced mathematics option in high school. It is also strongly recommended that students have taken the Calculus course in semester 1 or, at a minimum, have a basic knowledge of polynomials and complex numbers.

1 continuous assessment (written) + 1 final exam (written).

Content manager: laurent.guieu@umontpellier.fr