AR/0022 - GEOMETRY
Academic Year 2022/2023
Free text for the University
ANDREA RATTO (Tit.)
- Teaching style
- Lingua Insegnamento
|[80/71] ARCHITECTURAL SCIENCE||[71/00 - Ord. 2017] PERCORSO COMUNE||5||50|
The aim of the course is to provide students with the practical knowledge which is necessary to be able to work independently on various topics of basic mathematics.
In this course, we develop in parallel topics in geometry and linear algebra.
(I) Knowledge: The student must acquire the mathematical knowledge and methodologies that will enable him to study, critically and independently, scientific texts in all the areas related to engineering sciences.
(II) Skills (to know how to do): Although this is a basic course, the lecturer points out that the logical-deductive skills developed through the course will find significant application in every field of professional activity.
(III) Behaviour: The mental effort and concentration required to tackle the course are important in the formation of a modus vivendi respectful of others, thus promoting the maturation of interpersonal skills and sense of responsibility.
(IV) Communications skills: the student will achieve the skill to communicate what he (she) learnt. Moreover, he (she) will be able to justify the choice of a specific method or tool for the solution of a problem concerning linear algebra and analytic geometry.
(V) Skills to learn: thanks to the notions and methods acquired by means of this course, the Student will be able to progress further in the study of the applications of mathematics to problems in all the fields of architecture and engineering.
By way of summary:
1) At the end of the course, Students are expected to know the basic tools of linear algebra and analytic geometry.
2) They must understand the geometric and numerical meaning of linear algebra in order to use it in other scientific areas.
3)They must grasp the logical structure of the mathematical language in order to be able to study any scientific text.
4) The course should prove useful both to increase general scientific culture and to form the skills which are necessary to interact with other scientific disciplines.
Preliminary mathematical knowledge required: elementary arithmetic and algebraic operations (necessary). Basic notions about Euclidean geometry (important).
The first week is devoted to the illustration of the required preliminary mathematical knowledge to address the topics of the course. Therefore, there is no required specific prior knowledge. On the other hand, the course requires that the student possess, at least up to a reasonable extent, the skill of understanding verbal and logical reasoning. More specifically, students with special initial difficulties can refer to the book: Matematica: 2^3 capitoli per tutti" shown below.
§0 (5 hours). Basic notions in set theory, algebraic and polynomial methods in calculus, basic trigonometry.
§1 (4 hours). Vectors: operations with vectors, scalar product, wedge product and mixed product.
§2 (9 hours). Analytic geometry in 2 and 3 dimensions: straight lines, planes, projections, angles, distances, spheres; conics in canonical form.
§3 (5 hours). Real and complex numbers: basic properties, exercises and applications.
GUIDED EXERCISE ON TOPICS §§0--3: 3 hours.
§4 (7 hours). Matrices: operations with matrices, determinants, rank, inverse matrix. Elementary transformations on rows and columns and their effects on rank and determinants. Rotations, translations,theory of conics.
§5 (2 hours). Quick overview on vector spaces: linear independence, bases, orthonormal bases.
§6 (6 hours). Linear systems: Theorem of Rouché-Capelli, solving methods.
§7 (7 hours). Diagonalization of matrices: eigenvalues and eigenvectors, characteristic polynomial, methods of diagonalization. Applications to the study of conics. Quadrics.
GUIDED EXERCISE ON TOPICS §§4--7: 2 hours.
Traditional lessons (streaming available) at the blackboard, possibly with the help of projections of digital slides on a screen. Each lesson begins with a brief summary of the concepts covered in previous lessons and ends with the assignment of homework. The course is also supported by tutorial activity addressed to improve the ability to solve mathematical questions, and specifically conceived for the preparation of the written examination. In order to achieve the skill to apply knowledge, two guided practical sessions of 3 hours each are organized within the 50 hours of the course (constructive assessment). The first exercise session deals with parts §§0-3 of the program, while the second one concerns the remaining topics.
Verification of learning
Written test on the entire program. There will be 2 written tests (at half and at the end of the semester respectively) that enable the Student to pass the exam right at the end of the course. ALL the written tests consist in solving a series of problems, each of which is assigned a score. The score is determined taking into account:
1) The method of solution;
2) The accuracy in the computations;
3) The clarity of the exposition.
The student, solving the proposed problems, should demonstrate that he (she) has achieved the skills which are necessary to work with the methods and theories of linear algebra and geometry treated in the course.
The maximum total score is 33. The mark 30 cum laude is attributed to Students which obtain a final score greater or equal to 32.
Details about this type of problems are provided extensively during the lessons and in the reference textbook. Past exam papers are available on the website of Andrea Ratto.
Due to restrictions imposed by the general security policies of the university in connection with the evolution of the Covid-19 pandemic, it is possible that the exam will be carried out via the online platform Microsoft Teams. Questions and evaluation will be consistent with what has been described above.
1) Strang, Gilbert. Introduction to Linear Algebra. 4th ed. Wellesley, Wellesley-Cambridge Press, February 2009. ISBN: 9780980232714.
2) M.R. Spiegel, S. Lipschutz, D. Spellman (2009). Vector Analysis (Schaum’s Outlines) (2nd ed.). McGraw Hill. ISBN 978-0-07-161545-7.
Further information, the slides of the lectures and past papers are available on the web page of the Lecturer.