20355 - MATHEMATICS - PREPARATORY COURSE
Department of Decision Sciences
FEDERICO MARIO GIOVANNI VEGNI
Class 1: PAOLO LEONETTI: FEDERICO MARIO GIOVANNI VEGNI, Class 2: FEDERICO MARIO GIOVANNI VEGNI
Synchronous Blended: Lessons in synchronous mode in the classroom (for a maximum of one hour per credit in remote mode)
Suggested background knowledge
Mission & Content Summary
MISSION
CONTENT SUMMARY
Linear algebra:
- Euclidean spaces: geometric and algebraic approaches. Vectors in R^n. Operations with vectors. Matrices. Linear Space: linear dependance and independance. Dimension and bases of the linear space. Examples. Straight lines and planes in R^3. Linear systems: structure of solutions. Linear functions between euclidean spaces. Representation theorem. Eigenvalues and eigenvectors of a linear transformation. Spectral theorem for symmetric matrices.
Quadratic forms:
- Definitions and applications. Examples.
Curves in the plane and in the space:
- Straight lines in space. Parametric representation of a trajectory. Speed and tangent vector.
Functions in several variables:
- Level lines and contour map. Partial derivatives, gradient. Differential. Higher order derivatives. Derivative of a composite function. Hessian matrix. Implicit functions. Implicit function theorem. Jacobian matrix.
Optimization problems:
- Unconstrained optimization. The first order sufficient conditions. Fermat's theorem. Taylor polynomial of order two. Concavity and convexity. Second order sufficient conditions. Local-global theorem. Constrained oprimization. Lagrange multipliers technique. Meaning of multipliers.
Intended Learning Outcomes (ILO)
KNOWLEDGE AND UNDERSTANDING
At the end of the course student will be able to...
— Carry out a formal mathematical proof.
— Recognize the abstract mathematical structures that underlie modern economic theories.
— Master operations on functions and vectors.
APPLYING KNOWLEDGE AND UNDERSTANDING
At the end of the course student will be able to...
— Apply to economics and to the social sciences the basics of mathematics.
— Work out both the quantitative and the qualitative perspectives.
Teaching methods
- Face-to-face lectures
- Exercises (exercises, database, software etc.)
DETAILS
Face to face lessons have the aim of involving students in a rapid and effective review course.
Assessment methods
Continuous assessment | Partial exams | General exam | |
---|---|---|---|
|
x |
ATTENDING AND NOT ATTENDING STUDENTS
This preparatory course does not include a final exam. A continuous assessment is carried out stimulating the students’ engagement during face to face lectures.
Teaching materials
ATTENDING AND NOT ATTENDING STUDENTS
Lecture notes
FEDERICO MARIO GIOVANNI VEGNI
Synchronous Blended: Lessons in synchronous mode in the classroom (for a maximum of one hour per credit in remote mode)
Suggested background knowledge
Mission & Content Summary
MISSION
CONTENT SUMMARY
Linear algebra:
- Euclidean spaces: geometric and algebraic approaches. Vectors in R^n. Operations with vectors. Matrices. Linear Space: linear dependance and independance. Dimension and bases of the linear space. Examples. Straight lines and planes in R^3. Linear systems: structure of solutions. Linear functions between euclidean spaces. Representation theorem. Eigenvalues and eigenvectors of a linear transformation. Spectral theorem for symmetric matrices.
Quadratic forms:
- Definitions and applications. Examples.
Functions in several variables:
- Level lines and contour map. Partial derivatives, gradient. Differential. Higher order derivatives. Derivative of a composite function. Hessian matrix. Jacobian matrix.
Optimization problems:
- Unconstrained optimization. The first order sufficient conditions. Fermat's theorem. Taylor polynomial of order two. Concavity and convexity. Second order sufficient conditions. Local-global theorem. Constrained oprimization. Lagrange multipliers technique. Meaning of multipliers.
Intended Learning Outcomes (ILO)
KNOWLEDGE AND UNDERSTANDING
At the end of the course student will be able to...
— Carry out a formal mathematical proof.
— Recognize the abstract mathematical structures that underlie modern economic theories.
— Master operations on functions and vectors.
APPLYING KNOWLEDGE AND UNDERSTANDING
At the end of the course student will be able to...
— Apply to economics and to the social sciences the basics of mathematics.
— Work out both the quantitative and the qualitative perspectives.
Teaching methods
- Face-to-face lectures
DETAILS
Face to face lessons have the aim of involving students in a rapid and effective review course.
Assessment methods
Continuous assessment | Partial exams | General exam | |
---|---|---|---|
|
x |
ATTENDING AND NOT ATTENDING STUDENTS
This preparatory course does not include a final exam. A continuous assessment is carried out stimulating the students’ engagement during face to face lectures.
Teaching materials
ATTENDING AND NOT ATTENDING STUDENTS
Lecture notes