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.
