Algebra and geometry
The aim of this course is to learn analitic algebra and analitic geometry topics. A further aim is to apply these techniques in other scientific disciplines.
Learning outcomes (declined compared with the Dublin descriptors)
-Knowledge and understanding. Knowledge of analitic algebra and analitic geometry topics. The student will be able to state and prove basic theorems.
-Ability to apply knowledge and understanding. The ability to understand the problems proposed during the course, the ability to correctly apply the theoretical knowledge. The student will be able to study of the graphs of elementary functions, to solve integration problems of elementary character, to discuss the nature of numerical sequences and series.
-Making judgments. Develop the ability to critically evaluate the problems and propose the most appropriate approach.
-Communication skills. Ability to report and present the results with a logical-deductive and synthetic exposition
-Ability to learn.Ability to develop, outline, summarize the contents.
Elementary Algebra. Elements of Euclidean geometry. Elements of analytic geometry in the plane. First elements of mathematical logic : concepts , theorem , demonstration , role of examples and counterexamples.
Sets, operations, applications, algebraic structures: group ring field. (1 ECTS) Linear Algebra Vectors Matrices- Linear Systems. Vector spaces on R. Internal and external operations. Subspaces. Subspaces generated by a sequences of vectors. Linear dependence and independence-independent systems - Basis and dimension of a vector space. Changes reference- (1 ECTS) Matrices Determinant of a square matrix and its property- Rank of a matrix- Invertible Matrices. Cramer's rule for solving linear systems, linear representation of subspaces of R^n by using linear systems. (1 ECTS) Liner maps, definition and first properties. Kernel and Immage of a liner map.
Isomorphisms between vector spaces. Matrices and linear applications. Endomorphisms and isomorphisms- Diagonalization of endomorphisms and matrices- definitions and properties - characterizations of endomorphisms and diagonalization matrices - Isomorphism and coordinated representation of subspaces of a vector space by any linear systems in a given frame. (1 ECTS) Analytic geometry in the plane and space Linear dependence in the plane and in the space of geometric vector. Inner product standard- Orthogonal frames. Cartesian orthogonal monometric frame in spaces- Changes of frames. Representation of the line in the space- Direction cosines of a directed line. intersection of two lines and parallelism conditions. Orthogonality between lines. Midpoint and axis of a segment. (1 ECTS) Monometric frame in Cartesian space-Changes of frames- Vector product in the space of geometric vectors. Representation of line-parallelism and orthogonality between planes- Representation of the line in space-Directions a line - Pencil of planes- parallelism and othogonality between lines. Orthogonality and parallelism between lines and planes. Midpoint
- Algebraic structures (4 hours)
- Linear algebra (20 hours)
- Linear applications (10 hours)
- Analytic geometry in the plane and space (14 hours)
Lectures, exercise sessions and homeworks.
- J. S. Robinson, A Course in LINEAR ALGEBRA with Applications, World Scientific.
- K. W. Gruenberg A.J. Weir, Linear Geometry, Springer- Verlgar New York
The objective of the exam is to check the level of achievement of the above-mentioned training objectives.
The exam is divided into two parts:
- a written test that aims to evaluate the ability to correctly use the theoretical knowledge acquired during the course to solve mathematical problems. The student who does not show sufficient mastery of the arguments is not admitted to the next test. The expected time is 2 hours.
- an oral test in which the ability to link and compare different aspects of the course will be evaluated.
The final vote takes into account the evaluation of both tests.
Lectures are in Italian. The professor is fluent in English and is available to interact with students in English, also during the examination.
- Wednesday 11.15 - 13.15 (Betta)
- Friday 13:00-15:00 (Feo)