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Course 2023-2024 a.y.

20355 - MATHEMATICS - PREPARATORY COURSE

Department of Decision Sciences

Course taught in English

Go to class group/s: 1 - 2

DES-ESS (I sem. - P) - DSBA (I sem. - P)

Classes: 1 (I sem.) - 2 (I sem.)


Synchronous Blended: Lezioni erogate in modalità sincrona in aula (max 1 ora per credito online sincrona)

Suggested background knowledge

For a fruitful and effective learning experience, it is recommended a preliminary knowledge of basic calculus (limits, derivatives, integrals), of vector spaces, of linear maps and of matrix calculus.


Mission & Content Summary
MISSION

Deep familiarity with some topics generally carried out in the first-level courses in Mathematics is essential for a good understanding of the contents of the Advanced Mathematics for Economics and Social Sciences Course. These arguments are practically and theoretically reviewed along the preparatory Course, combining the analytical approach to the geometrical aspects and focusing on the economic interpretation.

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...

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...

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
  • Oral individual 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

    Last change 04/07/2023 17:58

    EMIT (I sem. - P) - PPA (I sem. - P) - TS (I sem. - P)

    Classes: 2 (I sem.)
    Instructors:
    Class 2: DOVID FEIN


    Synchronous Blended: Lezioni erogate in modalità sincrona in aula (max 1 ora per credito online sincrona)

    Suggested background knowledge

    For a fruitful and effective learning experience, it is recommended a preliminary knowledge of basic calculus (limits, derivatives, integrals), of vector spaces, of linear maps and of matrix calculus.


    Mission & Content Summary
    MISSION

    Deep familiarity with some topics generally carried out in the first-level courses in Mathematics is essential for a good understanding of the contents of the Advanced Mathematics for Economics and Social Sciences Course. These arguments are practically and theoretically reviewed along the preparatory Course, combining the analytical approach to the geometrical aspects and focusing on the economic interpretation.

    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...

    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...

    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
  • Oral individual 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

    Last change 06/06/2023 13:21