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## Program

EN IT

### 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:

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.