Course 2024-2025 a.y.

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.

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.

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.

• Practical Exercises

Face to face lessons have the aim of involving students in a rapid and effective review course.

This preparatory course does not include a final exam. A continuous assessment is carried out stimulating the students’ engagement during face to face lectures.

Lecture notes

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.

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.

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.

• Lectures

Face to face lessons have the aim of involving students in a rapid and effective review course.

This preparatory course does not include a final exam. A continuous assessment is carried out stimulating the students’ engagement during face to face lectures.

Lecture notes