20355  MATHEMATICS  PREPARATORY COURSE
Department of Decision Sciences
FEDERICO MARIO GIOVANNI VEGNI
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. Localglobal 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
 Practical Exercises
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
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. Localglobal 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
 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