Aggiornato A.A. 2024-2025
Prerequisites:
All the material covered in Mathematics 1
Program:
Integral Calculus
Definition of antiderivative and integral of a function.
Integrals of elementary functions.
Integration by substitution, Integration by parts, integrals of ratio of polynomials.
Construction of the Riemann Integral. Properties of Riemann Integral, Definite integral, Fundamental Theorem of Calculus (FTC) with proof. Applications of the FTC.
Computation of areas. Improper Integrals. Mean Value Theorem.
Linear Algebra
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 (with proof).
Matrix-vector product, product between matrices, Properties of the matrix product. Matrix product is not commutative: examples. Definition of the inverse of a matrix.
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 if 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.
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 linear 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.
Linear dependence and linear independence of vectors.
Rank of a matrix. Methods for computing the rank of a matrix: Gauss Elimination algorithmand Kronecker Theorem.
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.
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, computation.
Optimization
Functions of two variables: domain, level curves, partial derivatives, gradient. Stationary points. Hessian matrix, concavity, convexity. Tangent plane, Optimization of functions of two variables.
Bibliography and Teaching material:
Knut Sydsæter, Peter Hammond, Arne StrØm, Essential Mathematics for Economic Analysis
Carl P. Simon, Lawrence Blume, Mathematics for Economists
Slides and Exercises from previous years https://economia.uniroma2.it/ba/business-administration-economics/corso/materiali/1525/
Teaching methods:
Theoretical lectures and exercise classes
Exam Rules:
Mathematics is a single corse split in two modules. After the final exam only one grade will be registered.
The exam consists of a set of exercises and theoretical questions on the material covered during the course.
The exam is passed if the mark is grater than or equal to 18/30.
Prof.ssa Lucia Leonelli
