Mathématiques II
Course Description
Lectures and exercise sessions for undergraduate mathematics covering linear algebra, optimization, and regression.
Theory Sheets
1. Matrices
Definition of matrices, basic operations (addition, multiplication), matrix rank, and elementary row transformations.
2. Determinants
Computing determinants, properties of determinants, and their geometric interpretation as signed volumes.
3. Matrix Inversion
Finding inverse matrices using cofactors and elementary transformations, conditions for invertibility.
4. Linear Transformations
Linear forms, linear transformations as matrices, kernel and image, solving linear systems.
5. Eigenvalues and Eigenvectors
Definition and computation of eigenvalues via characteristic polynomials, finding eigenvectors, eigenspaces and their properties.
6. Quadratic Forms
Quadratic forms and their classification (positive/negative definite, indefinite), Sylvester's law of inertia, connection to eigenvalues.
8. Leontief Model
Input-output economic model, trade coefficients, computing production levels from export demands, trade balance.
9. Functions of Many Variables
Partial derivatives, total differential, gradient, elasticity, and higher-order partial derivatives.
10. Second Order Differential
Hessian matrix, Taylor expansions, first and second order conditions for free (unconstrained) extrema.
11. Linear Regression
Least squares method, normal equations, gradient and Hessian of quadratic functions, generalized and polynomial regression.