6 CFU (teacher: Andrey Sarychev): Real numbers. Function of one variable. Limits. Derivatives. Taylor polinomial. Functions of several variables. Partial derivatives and applications. Complex numbers.
- 3 CFU (teacher Francesco Rosai): Linear systems: Gaussian Elimination. Vector spaces. Matrices and determinants. Linear maps. Analytic Geometry of plane and Space. Diagonalization of a matrix
E.Giusti, Elementi di Analisi Matematica. Bollati Boringhieri, 2008 e posteriori.
G.Anichini, G.Conti, Geometria analitica e algebra lineare, Pearson, Milano, 2009.
Learning Objectives
Knowledge acquired:
The course will provide knowledge of the concepts and results regarding real and complex numbers, functions of one and several real variables. The course will also provide knowledge of Linear Algebra and of Analytic Geometry of Space and Plane.
Competence acquired
The student will be able to compute limits and derivatives. By means of these tools he will be able to study the properties of functions of one or several variables. The student will also be able to study and solve linear systems; to deal with vector spaces or subspaces and with linear maps; to find eigenvalues and eigenvectors of a linear map; to write equations of lines and planes; to use the notion of a distance for analytic representation of geometric objects.
Skills acquired (at the end of the course):
The student will acquire the ability of studying and solving linear and nonlinear equations and inequalities and to solve optimization problems with one or several variables. The student will also acquire the ability to use linear system theory and Gauss algorithm in several applications; he will be able to use geometric vectors to solve problems in analytic geometry of space; will be able to identify diagonalizzability of a matrix.
Prerequisites
Courses to be used as requirements (required and/or recommended)
None.
Courses required: None.
Courses recommended: None.
Teaching Methods
Lectures
Type of Assessment
Exam modality: written examination, oral discussion of the written examination
Course program
Recapitulation on equations and inequalities. Numbers: natural, integers, rational. Real numbers, properties. Axiom of completion. Elementary functions: linear functions, absolute value, power, exponential, logarithm, trigonometric functions and their inverse. Properties and computation of limits. Comparison theorem, sign persistence, product of limited by infinitesimal function. Continuous functions. Theorem of zeroes, bisection method. Weierstrass Theorem. Derivatives, computation of derivatives. Increasing and decreasing functions. Extremal points. Convexity and concavity. Graph of a function. Taylor polynomial. Taylor formula in the computation of limits. Functions of several variables: limits, continuity. Partial derivatives, gradient, Schwarz Theorem. Local maxima and minima, Hessian. Differentiability.
Complex numbers and operations with them: sum, product, conjugation, division, power, radicals.
Geometric vectors. Vector addition and scalar multiplication and characteristic properties. Linear combinations, linear dependence and independence. Parallel and coplanar vectors and equivalent conditions. Theorem Representation of a vector as a linear combination of three linearly independent; uniqueness. Inner, cross and mixed product and their properties. Computation of the products in an orthonormal basis.
Vector spaces: definitions, properties, examples.
Linear systems. Gaussian Elimination. Consistent systems and characteristic properties. Rouché-Capelli Theorem.
Matrix multiplication and characteristic properties.
Invertible matrices, their properties and computation.
Construction of a basis from a set of generators by means of Gaussian Elimination.
Linear and affine subspaces.
Determinant of a square matrix and characteristic properties. The determinant of a non-singular matrix. Kramer Theorem.
Linear maps. Injective, surjective,bijective linear maps. The linear map defined by a matrix. Kernel, image and the rank-nullity theorem.
Affine Geometry: Cartesian coordinate systems for a line, a plane and a three-dimensional space. Parametric and Cartesian equations of the line and planes; conditions for line-line and line-plane intersection or parallelism.
Elements of Euclidean Geometry: distance formulae and orthogonality conditions.
Eigenvalues and eigenvectors of a linear map. Analytic determination of eigenvalues. The real and the complex case. Diagonalization of a matrix; the case of matrices with distinct eigenvalues. The general case: algebraic and geometric multiplicities.