## Program

### Updated A.Y. 2022-2023

**Calculus**

- Integral Calculus. Definite and indefinite integrals, Integral properties, The fundamental theorem of calculus, Integration by parts and integration by substitution, Improper integrals.

**Linear Algebra**

- Linear Spaces. The algebra of vectors, Euclidean Spaces, Inner product, Linear independence.

- Matrices. Matrix algebra, Determinant, Inverse Matrix.

- Systems of Linear equations. The Gauss Elimination Algorithm, Rouche Capelli theorem.

- Eigenvalues and eigenvectors, Diagonalization.

**Optimization**

- Calculus of several variables. Domain, Partial derivatives, gradient, hessian matrix. Stationary points. Countour curves.

- Unconstrained optimization. First and second order conditions

- Constrained optimization with Equality constraints. First and second order conditions

- Optimization on a set

**Suggested Books**:

Essential Mathematics for Economic Analysis

Carl P. Simon, Lawrence Blume, Mathematics for Economists

**Teaching Material **

**Available on the webpage of the course and on Teams:**

- Slides of the course
- Additional Exercises

*The final Syllabus will be available only at the end of the course.*

*You can find below a detailed program which will be update on a daily basis during the teaching period.*

Lecture 1 (31/10/2022): Definition of atiderivative and integral of a function. Integrals of elementary functions

Lecture 2 (2/11/2022): Integration by substitution and Integration by parts

Practice 1 (3/11/2022): Integrals (elementary integrals, integration by substitution and integration by parts)

Lecture 3 (4/11/2022): Exercises on Integration by parts and Substitution. Construction of the Riemann Integral

Lecture 4 (7/11/2022): Properties of Riemann Integral, Definite integral, Fundamental Theorem of Calculus (FTC). Applications of the FTC.

Lecture 5 (8/11/2022): Exercises on definite integrals, application of FTC, computation of areas. Improper Integrals. Mean Value Theorem.

Lecture 6 (9/11/2022): Matrices and Vectors: definitions, operations with matrices and vectors (sum, subtraction, multiplication by a real number, transpose, inner product). Geometric interpretation of vectors. Linear combination of vectors. Any vector in R^n is a linear combination of standard vectors.

Practice 2 (10/11/2022): Definite Integrals, applications of FTC, Computation of areas. Matrices.

Lecture 7 (14/11/2022): Matrix-vector product, product between matrices, Properties of the matrix product. Matrix product is not commutative: examples. Definition of the inverse of a matrix.

Lecture 8 (15/11/2022): Theorem: the inverse of a matrix is unique (with proof). Inverse of a 2x2 matrix. Determinant of a matrix (cofactor expansion by row and by column). Properties of determinants. Theorem: A matrix is invertible iff its determinant is not equal to zero. Properties of the inverse of a matrix (with proof). The cofactor matrix. Computation of the inverse of a matrix. Special matrices: Diagonal and Triangular matrices.

Lecture 9 (16/11/2022): Definition of a linear equation and a system of linear equations. Matrix-vector expression for a system of linear equations. Definition of the solution of a system of liunear equations. Consistent and Inconsisten systems. Theorem: A homogeneous system is always consistent (with proof). Elementary Row operations and the Gauss Elimination Algorithm. Row Echelon Form, consistency and inconsistency of a system.

Practice 3: Product of matrices, determinant of a matrix, Inverse of a matrix, linear systems.

Lecture 10 (21/11/2022): Exercises on linear systems, linear dependence and linear independence of vectors.

Lecture 11 (22/11/2022): Linear independence of vectors, rank of a matrix. Methods for computing the rank of a matrix: Gauss Elimination algorithm and Kronecker Theorem.

Lecture 12 (23/11/2022): Exercises on the computation of the rank. Proposition: Given a set of vectors v_1, ..., v_n in R^k and a vector b in R^k which is a linear combination of vectors v_1, ..., v_n, then the matrix V and the matrix tilde V have the same rank. (with proof). Rouché-Capelli Theorem (Statement and interpretation). Parametric systems.

Practice 4 (24/11/2022): Linearly dependent/independent vectors, rank, parametric systems

Lecture 13 (28/11/2022): geometric interpretation of the solutions of a linear system; cartesian and parametric equation of a line through two points, and through a point in a direction v; parametric equation of a plane through a point and parallelt to two vectors, point-normal equation of a plane in R^3. Eigenvalues and Eigenvectors definition.

Lecture 14 (29/11/2022): Eigenvalues and Eigenvectors computation; functions of two variables: domain.

Lecture 15 (30/11/2022): Functions of two variables: level curves, partial derivatives, gradient.

Practice 5 (1/12/2022): Parametric/cartesian equations of lines/planes. Eigenvalues and Eigenvectors. Functions of two variables: domain, level curves, gradient.

Lecture 16 (5/12/2022): Partial derivatives, gradient, Stationary points. Hessian matrix, concavity, convexity.

Lecture 17 (6/12/2022): Tangent plane, Optimization of functions of two variables.

Lecture 18 (7/12/2022): Again on Optimization of functions of two variables.

Practice 6 (7/12/2022): Tangent plane and Optimization of functios of two variables.