Syllabus
Prerequisiti
Prerequisites
Programma
Definizione di Primitiva e integrale di una funzione. Integrali di funzioni elementari.
Integrazione per sostituzione, per parti e del rapporto di polinomi.
Costruzione dell'integrale di Riemann. Proprietà dell'integrale di Riemann, Integrale definito, Teorema Fondamentale del Calcolo (con dimostrazione). Applicazioni del TFC
Calcolo di Aree. Integrale Improprio. Teorema del Valore Medio.
Algebra Lineare
Matrici e Vettori: definizione, operazioni con matrici e vettori, interpretazione geometrica diveettori, combinazioni lineari di vettori. Ogni vettore in R^n è una combinazione lineare di vettori standard (con dimostrazione).
Prodotto Matrice-vettore, prodotto tra matrici. Proprietà del prodotto tra matrici. Il prodotto tra matrici non è commutativo (esempi). Inversa di una matrice.
Unicità dell'inversa di una matrice (con dim.) Inversa di una matrice 2x2. Determinante di una matrice (con i cofattori). Proprietà dei determinanti.
Teorma: una matrice è invertibile se e solo se il suo determinante non è zero. Proprietà dell'inversa (con dimostrazione). Matrice dei cofattori. CAlcolo dell'inversa di una matrice.
Matrici speciali: diagonali e triangolari.
Definizione di equazione lineare e sistema di equazioni lineari. Espressione matriciale di unsistema di equazioni lineari. Soluzione di un sistema di eq uazioni lineari: consistenza, inconsistenza e numero di soluzioni. Teorema: un sistema omogeneo è sempre consistente (con dim.). Operazioni elementari per righe, forma a scalini.
Dipendenza e indipendenza lineare di verrori. Rango di una matrice. Metodi per il calcolo del rango: algoritmo di eliminazione di Gauss e teorema di Kronecker.
Proposizione: Dato un insieme di vettori v_1, ..., v_n in R^k e un vettore b in R^k che è combinazione lineare di v_1, ..., v_n, la matrice V e la matrice tilde V hanno lo stesso rango(con dim). Teorema di Rouché-Capelli (enunciato e interpretazione). Sistemi parametrici.
Interpretazione geometrica delle soluzioni di un sistema lineare; equazione parametrica e cartesiana di una retta per due punti; e per un punto nella direzione v; equazione parametrica del piano per un punto e parallelo a due vettori; equazione point-normal di un piano in R^3. Autovalori e Autovettori di una matrice: definizione e calcolo.
Ottimizzazione di funzioni di due variabili
Funzioni di due variabili: dominio, curve di livello, derivate parziali, gradiente, matrice Hessiana. Punti stazionari, concavità e convessità. Piano tangente e ottimizzazione di funzioni di due variabili.
Program
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.
Testi Adottati
Books
Bibliografia
Carl P. Simon, Lawrence Blume, Mathematics for Economists
Bibliography
Carl P. Simon, Lawrence Blume, Mathematics for Economists
Modalità di svolgimento
Teaching methods
Regolamento Esame
La prova scritta è formata da una serie di esercizi e domande teoriche sul programma analizzato durante in corso.
L'esame si ritiene superato se lo studente ottiene una votazione di almeno 18/30.
Ogni voto verrà registrato, inclusa la bocciatura.
E' possibile rifiutare un voto maggiore o uguale a 18/30 una sola volta e ripetere l'esame. In tal caso, al secondo tentativo il voto viene registrato.
Exam Rules
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.
Every result of the exam will be registered, including "fail". Students are allowed to reject a mark greater or equal to 18 only once and re-take the exam. In the case, at the second time the grade will be registered.
Updated A.Y. 2022-2023
Calcolo
- Calcolo integrale. Integrali definiti, proprietà degli integrali, Teorema Fondamentale del calcolo integrale. Metodi di risoluzione degli integrali: Integrazione per Parti e per Sostituzione. Integrali indefiniti.
Algebra Lineare
- Spazi lineari. Algebra dei vettori, Prodotto Interno, Indipendenza Lineare.
- Matrici. Algebra delle Matrici, Determinante, Rango, Matrice inversa, Autovalori e Autovettori.
- Sistemi di Equazioni lineari. Algoritmo di Gauss, Teorema di Rouche-Capelli.
Ottimizzazione
- Calcolo in più variabili. Dominio, Derivate Parziali, Gradiente, Matrice Hessiana. Punti Stazionari. Curve di Livello.
- Ottimizzazione non vincolata. Condizioni del primo e del secondo ordine.
- Ottimizzazione vincolata con vincoli di Uguaglianza. Condizioni del primo e del secondo ordine.
Testi consigliati:
Knut Sydsaeter, Peter Hammond and Arne Strom, Essential Mathematics for Economic Analysis
Carl P. Simon, Lawrence Blume, Mathematics for Economists
Materiale del corso
Disponibile sulla pagina web del corso
- Slides delle lezioni
- Esercizi
Il programma finale sarà disponibile solo a conclusione del corso.
Di seguito gli argomenti trattati lezione per lezione, aggiornati quotidianamente durante il semestre di insegnamento.
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 (24/22/2022): 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.
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:
Knut Sydsaeter, Peter Hammond and Arne Strom, 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.
Updated A.Y. 2020-2021
Calcolo
- Calcolo integrale. Integrali definiti, proprietà degli integrali, Teorema Fondamentale del calcolo integrale. Metodi di risoluzione degli integrali: Integrazione per Parti e per Sostituzione. Integrali indefiniti.
Algebra Lineare
- Spazi lineari. Algebra dei vettori, Spazi Euclidei, Prodotto Interno, Indipendenza Lineare.
- Matrici. Algebra delle Matrici, Determinante, Traccia, Rango, Matrice inversa, Autovalori e Autovettori.
- Sistemi di Equazioni lineari. Algoritmo di Gauss, Teorema di Rouche-Capelli.
Ottimizzazione
- Calcolo in più variabili. Dominio, Derivate Parziali, Gradiente, Matrice Hessiana. Punti Stazionari. Curve di Livello.
- Ottimizzazione non vincolata. Condizioni del primo e del secondo ordine.
- Ottimizzazione vincolata con vincoli di Uguaglianza. Condizioni del primo e del secondo ordine.
- Ottimizzazione su un insieme.
Testi consigliati:
Knut Sydsaeter, Peter Hammond and Arne Strom, Essential Mathematics for Economic Analysis
Carl P. Simon, Lawrence Blume, Mathematics for Economists
Materiale del corso
Disponibile sulla pagina web del corso e su Teams:
- Slides delle lezioni
- Esercizi aggiuntivi
Disponibile su Moodle:
- Self-check tests saranno pubblicati ogni settimana nella giornata di venerdì
Il programma finale sarà disponibile solo a conclusione del corso.
Di seguito gli argomenti trattati lezione per lezione, aggiornati quotidianamente durante il semestre di insegnamento.
Programma Dettagliato
22/02/2021 Antiderivative. Theorem: if a function has one antiderivative, then it has infinitely many antiderivatives (with proof). Indefinite integrals: definition. Integrals of Elementary Functions. Examples and Exercises.
23/02/2021 Integrals of Elementary Functions., Integration by substitution, integration by parts, Approximation of an area by Riemann sums. Examples and Exercises.
24/02/2021 Definite integral, Properties of definite integrals, The Fundamental Theorem of Calculus: statement and applications, The Mean Value Theorem: statement and geometric interpretation. Examples and Exercises.
01/03/2021 Improper integral. Matrix, Operations with matrices: sum and multiplication by a scalar (definition and properties). The null matrix. Examples and Exercises.
02/03/2021 The transpose of a matrix, Vectors, operations with vectors: sum, multiplication by a scalar, inner product (definition and properties). Linear combination of vectors. The standard vectors in R^n: definition and properties (with proof). Matrix-vector multiplication (definition and properties). The identity matrix: definition and property (with proof). Examples and Exercises.
03/03/2021 Matrix product (definition and properties), Matrix product is not commutative (two counterexamples), Diagonal, Lower Triangular and Upper Triangular matrices. Inverse of a matrix (definition), The inverse of a matrix is unique (with proof), Inverse of a 2x2 matrix (with proof), necessary and sufficient condition for invertibility, The determinant of a matrix: cofactor expansion by rows and by columns. Examples and Exercises.
08/03/2021 Properties of determinants. The determinant is not linear (example), Inverse of a matrix, Properties of the inverse of a matrix (proof of the property: (AP)^(-1)=P^(-1)A^(-1), the proof of the other properties are left as homework). Systems of linear Equations: definition of a linear system, Definition of the solution of a linear system, geometric interpretation of a linear system of two equations in two variables, The matrix form of a linear system, Homogeneous systems, Proposition: A homogeneous system is always consistent (with proof). Examples and Exercises.
09/03/2021 Elementary Row Operations, The Row Echelon form of a matrix, Solving a linear system via the Gauss Elimination Algorithm. Examples and Exercises.
10/03/2021 Linear systems: Examples and Exercises. Parametric Linear systems: Examples and Exercises. Linearly dependent vectors: definition.
12/03/2021 Linerly dependent and linearly independent vectors: Definitions. How to check if vectors are linearly dependent/independent: 1. Visual Inspection; 2. Using the definition. Rank of a matrix: definition and properties. Examples and Exercises.
15/03/2021 Computation of the rank of a matrix. Rouché-Capelli Theorem: statement and interpretation. Lines in R^n (parametric and Cartesian equation). Examples and Exercises.
16/03/2021 Planes and Hyperplanes in R^n. Geometric interpretation of the solutions of a linear system. Spanning sets: definition, examples. Proposition: The span of all standard vectors in R^n is R^n (with proof). How to check if a vector belongs to a Span. Examples and Exercises.
17/03/2021 Spanning sest: Dimension of a span, How to make the generating set smaller, basis of a linear space, Properties of the basis. Eigenvalues and Eigenvectors of a matrix: interpretation and definition. Proposition: An homogeneous system Tx=o has nontrivial solutions if and only is det(T)=0. How to use the proposition to compute Eigenvalues of a matrix A. Computation of Eigenvectors of a matrix A. Examples and Exercises.
19/03/2021 Eigenvalues and Eigenvectors of a matrix, Algebraic and Geometric multiplicity, Relationship between algebraic and geometric multiplicities (statement only), Diagonalization (necessary and sufficient conditions), Computation of Powers of matrices. Examples and Exercises.
22/03/2021 Diagonalization, Definiteness of a matrix. Examples and Exercises.
23/03/2021 Functions of two variables: Domain and Level curves. Examples and Exercises.
24/03/2021 Continuity of a function of two variables: definition, First Order Partial derivatives: definition and interpretation , Differentiability: definition, Gradient and stationary points: definition, geometric interpretation and tangent plane. Second Order Partial derivatives: definition. Schwarz Theorem (statement only). The Hessian matrix. Examples and Exercises.
29/03/2021 Twice continuously differentiable functions, Quadratic foms: matrix representation and the Hessian matrix. Unconstrained Optimization. Sedinition of global maxima and global minima, local maxima and local minima of functions of two variables Theorem: First order necessary condition for local maxima and local minima (only statement), Theorem: Second order sufficint condition for local maxima and local minima (only statement). Examples and Exercises.
30/03/2021 Unconstrained optimization: local maxima and local minima in case of semidefinite or null Hessian matrix. Globally concave and globally convex functions (Statement of the Theorem). Global maxima and Global minima: Theorem Second order sufficient condition (only statement). Constrained optimization with linear constraint. Examples and Exercises.
31/03/2021 More on Constrained optimization with linear constraint. Constrained optimization with nonlinear constraint: The Lagrange Multiplier Approach. Theorem: First order necessary condition (Statement and geometric interpretation in terms of level curves). Theorem: Second order sufficient condition. Examples and Exercises.
Updated A.Y. 2020-2021
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:
Knut Sydsaeter, Peter Hammond and Arne Strom, 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
Available on Moodle:
- Self-check tests are uploaded on Friday every week
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.
Detailed Program
22/02/2021 Antiderivative. Theorem: if a function has one antiderivative, then it has infinitely many antiderivatives (with proof). Indefinite integrals: definition. Integrals of Elementary Functions. Examples and Exercises.
23/02/2021 Integrals of Elementary Functions., Integration by substitution, integration by parts, Approximation of an area by Riemann sums. Examples and Exercises.
24/02/2021 Definite integral, Properties of definite integrals, The Fundamental Theorem of Calculus: statement and applications, The Mean Value Theorem: statement and geometric interpretation. Examples and Exercises.
01/03/2021 Improper integral. Matrix, Operations with matrices: sum and multiplication by a scalar (definition and properties). The null matrix. Examples and Exercises.
02/03/2021 The transpose of a matrix, Vectors, operations with vectors: sum, multiplication by a scalar, inner product (definition and properties). Linear combination of vectors. The standard vectors in R^n: definition and properties (with proof). Matrix-vector multiplication (definition and properties). The identity matrix: definition and property (with proof). Examples and Exercises.
03/03/2021 Matrix product (definition and properties), Matrix product is not commutative (two counterexamples), Diagonal, Lower Triangular and Upper Triangular matrices. Inverse of a matrix (definition), The inverse of a matrix is unique (with proof), Inverse of a 2x2 matrix (with proof), necessary and sufficient condition for invertibility, The determinant of a matrix: cofactor expansion by rows and by columns. Examples and Exercises.
08/03/2021 Properties of determinants. The determinant is not linear (example), Inverse of a matrix, Properties of the inverse of a matrix (proof of the property: (AP)^(-1)=P^(-1)A^(-1), the proof of the other properties are left as homework). Systems of linear Equations: definition of a linear system, Definition of the solution of a linear system, geometric interpretation of a linear system of two equations in two variables, The matrix form of a linear system, Homogeneous systems, Proposition: A homogeneous system is always consistent (with proof). Examples and Exercises.
09/03/2021 Elementary Row Operations, The Row Echelon form of a matrix, Solving a linear system via the Gauss Elimination Algorithm. Examples and Exercises.
10/03/2021 Linear systems: Examples and Exercises. Parametric Linear systems: Examples and Exercises. Linearly dependent vectors: definition.
12/03/2021 Linerly dependent and linearly independent vectors: Definitions. How to check if vectors are linearly dependent/independent: 1. Visual Inspection; 2. Using the definition. Rank of a matrix: definition and properties. Examples and Exercises.
15/03/2021 Computation of the rank of a matrix. Rouché-Capelli Theorem: statement and interpretation. Lines in R^n (parametric and Cartesian equation). Examples and Exercises.
16/03/2021 Planes and Hyperplanes in R^n. Geometric interpretation of the solutions of a linear system. Spanning sets: definition, examples. Proposition: The span of all standard vectors in R^n is R^n (with proof). How to check if a vector belongs to a Span. Examples and Exercises.
17/03/2021 Spanning sest: Dimension of a span, How to make the generating set smaller, basis of a linear space, Properties of the basis. Eigenvalues and Eigenvectors of a matrix: interpretation and definition. Proposition: An homogeneous system Tx=o has nontrivial solutions if and only is det(T)=0. How to use the proposition to compute Eigenvalues of a matrix A. Computation of Eigenvectors of a matrix A. Examples and Exercises.
19/03/2021 Eigenvalues and Eigenvectors of a matrix, Algebraic and Geometric multiplicity, Relationship between algebraic and geometric multiplicities (statement only), Diagonalization (necessary and sufficient conditions), Computation of Powers of matrices. Examples and Exercises.
22/03/2021 Diagonalization, Definiteness of a matrix. Examples and Exercises.
23/03/2021 Functions of two variables: Domain and Level curves. Examples and Exercises.
24/03/2021 Continuity of a function of two variables: definition, First Order Partial derivatives: definition and interpretation , Differentiability: definition, Gradient and stationary points: definition, geometric interpretation and tangent plane. Second Order Partial derivatives: definition. Schwarz Theorem (statement only). The Hessian matrix. Examples and Exercises.
29/03/2021 Twice continuously differentiable functions, Quadratic foms: matrix representation and the Hessian matrix. Unconstrained Optimization. Definition of global maxima and global minima, local maxima and local minima of functions of two variables. Theorem: First order necessary condition for local maxima and local minima (only statement), Theorem: Second order sufficient condition for local maxima and local minima (only statement). Examples and Exercises.
30/03/2021 Unconstrained optimization: local maxima and local minima in case of semidefinite or null Hessian matrix. Globally concave and globally convex functions (Statement of the Theorem). Global maxima and Global minima: Theorem Second order sufficient condition (only statement). Constrained optimization with linear constraint. Examples and Exercises.
31/03/2021 More on Constrained optimization with linear constraint. Constrained optimization with nonlinear constraint: The Lagrange Multiplier Approach. Theorem: First order necessary condition (Statement and geometric interpretation in terms of level curves). Theorem: Second order sufficient condition. Examples and Exercises.
Updated A.Y. 2019-2020
Calcolo
- Calcolo integrale. Integrali definiti, proprietà degli integrali, Teorema Fonìdamentale del calcolo integrale. Metodi di risoluzione degli integrali: Integrazione per Parti e per Sostituzione. Integrali indefiniti.
Algebra Lineare
- Spazi lineari. Algebra dei vettori, Spazi Euclidei, Prodotto Interno, Indipendenza Lineare.
- Matrici. Algebra delle Matrici, Determinante, Traccia, Rango, Matrice inversa, Autovalori e Autovettori.
- Sistemi di Equazioni lineari. Algoritmo di Gauss, Teorema di Rouche-Capelli.
Ottimizzazione
- Calculo in più variabili. Dominio, Derivate Parziali, Gradiente, Matrice Hessiana. Punti Stazionari. Curve di Livello.
- Ottimizzazione non vincolata. Condizioni del primo e del secondo ordine.
- Ottimizzazione vincolata con vincoli di Uguaglianza. Condizioni del primo e del secondo ordine.
- Ottimizzazione su un insieme.
Testo di riferimento: Carl P. Simon, Lawrence Blume, Mathematics for Economists
Il programma finale sarà disponibile solo a conclusione del corso.
Di seguito gli argomenti trattati lezione per lezione.
Programma delle lezioni:
17/02/2020 Integrali indefiniti, Integrali indefiniti elementari
18/02/2020 Soluzione di integrali indefiniti per sostituzione e integrazione per parti
20/02/2020 Integrali definiti, Teorema della media, Teorema fondamentale del calcolo, Integrali impropri
23/02/2020 Algebra lineare: Matrici e vettori. Operazioni con le matrici: somma e moltiplicazione per uno scalare e loro proprietà. Operazioni con vettori: somma e moltiplicazione per uno scalare e loro proprietà.
25/02/2020 Prodotto matrice-vettore e proprietà. Prodotto tra matrici e proprietà. Inversa di una matrice 2x2.
26/02/2020 Determinanti: espansione per cofattori. inversa di una matrice quadrata.
02/03/2020 Sistemi lineari: sistemi di 2 equazioni in 2 variabili, sistemi generali di equazioni lineari, Algoritmo di Gauss
04/03/2020 Esercizi: soluzione di sistemi con algoritmo di Gauss, dipendenza/indipendenza lineare
16/03/2020 Esercizi su sistemi parametrici, Esercizi su dipendenza/indipendenza lineare, Rango di una matrice, teorema di Rouché-Capelli
18/03/2020 Interpretazione del teorema di Rouché-Capelli, Applicazioni di sistemi lineari: il modello di Leontief. Vettori reali, prodotto interno, ortogonalità e parallelismo
19/03/2020 Rette, piani e iperpiani e interpretazione geometrica delle soluzioni di sistemi di equazioni lineari. Spazi generati da vettori.
23/03/2020 Spazi generati da vettori, basi, Autovalori e Autovettori
25/03/2020 Autovalori e Autovettori, Diagonalizzazione
26/03/2020 Diagonalizzazione, Funzioni di 2 Variabili: Dominio, Codominio, Curve di Livello
30/03/2020 Derivabilità di funzioni in 2 variabili, gradiente, matrice Hessiana, Funzioni di più di 2 variabili:dominio, gradiente e Matrice Hessiana
31/03/2020 Forme Quadratiche, Ottimizzazione di Forme quadratiche, Ottimizzazione non vincolata
01/04/2020 Ottimizzazione non vincolata, Condizioni del primo e del secondo ordine
02/04/2020 Ottimizzazione vincolata con vincoli di uguaglianza. Ottimizzazione su un insieme
Updated A.Y. 2019-2020
Calculus
- Integral Calculus. Definite and indefinite integrals, Integral properties, The fundamental theorem of calculus, Integration by parts and integration by substitution, Improper integralsLinear Algebra
Linear Algebra
- Linear Spaces. The algebra of vectors, Euclidean Spaces, Inner product, Linear independence.
- Matrices. Matrix algebra, Determinant, Trace, Rank, Inverse Matrix, Eigenvalues and eigenvectors, Diagonalization.
- Systems of Linear equations. The Gauss Elimination Algorithm, Rouche Capelli theorem.
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
Book: Carl P. Simon, Lawrence Blume, Mathematics for Economists
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.
Detailed Program:
17/02/2020 Indefinite Integrals, Elementary indefinite Integrals
18/02/2020 Solving Integrals by Substitution and Integration by parts
20/02/2020 Definite Integrals, The Mean Theorem, The Fundamental Thorem of Calculus, Improper Integrals
23/02/2020 Linear Algebra: Matrices, Vectors. Operations with matrices: sum and multiplication by a scalar and their properties. Operations with vectors: sum and multiplication by a scalar and their properties.
25/02/2020 Matrix-Vector product and properties. Matrix Product and properties. Inverse of a 2x2 matrix.
26/02/2020 Determinant: computation by cofactor expansion. Inverse of a square matrix
02/03/2020 Linear systems: System of two linear equations in two variables, General system of linear equations, The Gauss Algorithm
04/03/2020 Examples on the solution of systems with the Gauss Elimination Algorithm, Linear dependence/independence
16/03/2020 Examples on parametric systems, Examples on Linear dependence & Independence, Rank of a matrix, Rouché-Capelli Thorem
18/03/2020 Interpretation of Rouché-Capelli Theorem, Applications of Linear Systems: The Leontief model. Real Vectors, the inner product, Orthogonality and Parallelism
19/03/2020 Lines, Planes, Hyperplanes and the geometric interpretation of the solution of system of linear equations. Spanning sets
23/03/2020 Spanning Sets, Basis, Eigenvalues and Eigenvectors
25/03/2020 Eigenvalue and Eigenvectors, Diagonalization
26/03/2020 Diagonalization, Functions of two variables: Domain, Range, Level curves
30/03/2020 Differentiability, Gradient and Hessian Matrix, Functions of more that two variables: Domain, Differentiability, Gradient and Hessian Matrix
31/03/2020 Quadratic forms, Optimization of Quadratic forms, Unconstrained optimization
01/04/2020 Unconstrained Optimization: First and second order conditions
02/04/2020 Constrained optimization (with equality constraint). Optimization on a set.