60/64/172 - GEOMETRY 1
Academic Year 2021/2022
Free text for the University
BENIAMINO CAPPELLETTI MONTANO (Tit.)
- Teaching style
- Lingua Insegnamento
|[60/64] MATHEMATICS||[64/00 - Ord. 2017] PERCORSO COMUNE||8||64|
"Geometry 1" is the first course of a wide topic in Mathematics, known as "Geometry". In fact, the course deals with subjects - such as the study of vector spaces and linear maps - that play a fundamental role in all pure and applied Mathematics, and will therefore accompany the student throughout his or her future career.
The main goals of the course are described as follows.
KNOWLEDGE AND UNDERSTANDING ABILITIES
One among the aims of the course is to provide students with the basic elements of linear algebra, which will then be used in most of their subsequent studies.
The theoretical structure of the course consists of the development of the themes of the program, through the introduction of fundamental concepts and development of a series of theorems together with the correspondent proofs, accompanied by significant examples, exercises and applications.
In particular, the course aims to develop a rigorous mathematical language; assimilation of abstract concepts, algebraic structures, theorems and their proofs, related to linear algebra.
APPLIED KNOWLEDGE AND UNDERSTANDING ABILITIES
Learning of proof techniques; ability to solve new problems and standard exercises, in which you one has to develope a strategy and apply the concepts learned, or process a little proof similar to those seen in class. Using linear algebra tools in other contexts.
To assimilate the standard proof-techniques of linear algebra and to be able to use them autonomously in other contexts.
The student will be able to present and argue the solution of problems; he will also be able to discuss and properly show the most relevant results related to linear algebra.
TO LEARN SKILLS
Ability to learn to independently solve complex problems and exercises. Ability to read and understand ad undergraduate level textbooks of Mathematics.
There are no special prerequisites to profitably attend the course of Geometry 1. A new theory - Linear Algebra - will be illustrated. Thus, while it generalizes some topics touched in higher schools, it does not need specific prerequisites.
However, it may be useful to be familiar with some of the notions usually dealt with in higher schools, such as: arithmetic operations and elementary properties of numbers; numerical sets (natural, integer, rational, real); polynomials in one variable; resolution of polynomial equations and inequalities; the geometric meaning of the sine, cosine and tangent functions, and the main trigonometric formulas.
MATRICES, DETERMINANTS, LINEAR SYSTEMS
Matrices. Sum of matrices and product of a matrix by a scalar. Product of matrices. Determinants. Laplace theorems. Inverse matrix. Linear systems. Theorem of Cramer. The linear group.
Introductory examples: free vectors. Definition of vector space. Examples of vector spaces. Linear subspaces. Linear combinations and linear dependence of vectors. Generators systems. Finitely generated vector spaces. Base of a vector space. Rank of a matrix. Theorem of Rouché-Capelli.
Linear maps. Kernel and image of a linear map. Dimensional equation. The fundamental theorem of linear maps. Matrix associated of a linear map. The space L (V, W) of all linear maps between vector spaces V and W. Dual and bidual space. Dual base. Eigenvalues and eigenvectors. Diagonizable endomorphisms. Similar matrices. Spectral Theorem.
The course consists of 64 hours frontal lessons. Compatibly with the mixed teaching method foreseen in the Manifesto Accademico 2021-22 as a consequence of the COVID-19 emergency, the tools used for the lectures can be both the blackboard and tablet with projection system via classroom screen and via internet streaming.
The lessons schedule is indicated on the Math website
Students are also offered a 32-hours tutoring program, which includes a weekly meeting between students and a tutor, in order to check the exercises assigned by the teacher or to review some exercises where students encountered difficulties. These meetings are optional (but recommended) and will follow a timetable communicated by the teacher at the beginning of the course, and published on the course web page.
Moreover, at the this page students can find any information on the course, any notes of the teacher and the weekly exercises to do at home and
corrected by the tutor during the tutor's class.
Verification of learning
The exam consists in a written test and an oral examination, whose aims are checking the level of achievement of the aforementiones formative goals.
The written test takes 3 hours and consists of 3 or 4 exercises, similar to those described during the course. The aims of the written test is to ensure the correct understanding of the notions, the capability of find connections between them, and the problem solving skills of the student.
Students who pass the written test are admitted to the oral test at the blackboard. The oral test generally takes 40 minutes and consists in 3 questions about different topics of the course. It is passed if the student answers correctly to at least two questions on different topics of the course program. To do this, the student, during the oral exam, must write at the blackboard all the steps he is following to arrive at the conclusion of a reasoning. Exposure skills and the ability to relate concepts will also be evaluated. In any case, a too inadequate answer on a fundamental topic (for instance, linear indipendence of vectors) can affect the entire oral examination.
If the student does not pass the oral examination, he or she can repeat it once, without passing a new written test.
The final grade is determined by the assesstment of both tests.
The exams calendar is established by the Council of professors of the Mathematics course and it is published in the Math course website
Textbook: notes written by the professor, that can be found at his istitutional home-page
- M.R.Casali, C. Gagliardi, L. Grasselli - Geometria - Esculapio Editore
- L.Grasselli, C.Landi - Algebra lineare e Geometria, Esercizi risolti e commentati - Esculapio Editore
During the first lesson the objectives, the contents, the verification methods of the course will be described. Moreover, the web-page of the course, where students can find educational material (such as information, exercises, examination tests, etc.) will be presented.
The teacher devotes one day per week to receive students in his office for explanations or discussions. Such day will be announced announced at the beginning of the course and clearly indicated in the teacher's web-page. The teacher is also available to provide information to the students vi email (email@example.com) or phone (070/675-8520).
Cagliari University provides support for students with specific learning disability (SLD). Those interested can find more informations at this link: