
Course unit
TOPICS IN LINEAR ALGEBRA AND GEOMETRY
IN08122537, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2019/20
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Basic courses 
MAT/03 
Geometry 
9.0 
Course unit organization
Period 
First semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Lecture 
9.0 
72 
153.0 
No turn 
Prerequisites:

None. 
Target skills and knowledge:

The course aims to introduce familiarity with mathematical structures whose knowledge is indispensable in subsequent mathematics courses and in all engineering disciplines in which one uses matrices, linear functions, coordinates, complex numbers. In particular, the knowledge of the main theoretical aspects concerning complex numbers, vector spaces, linear functions and matrices as well as their applications in geometry, are expected. Students must achieve the ability to solve exercises and simple problems on all the aforementioned topics. 
Examination methods:

The verification of knowledge and expected skills is carried out with an exam divided into written and oral ones. In turn, the written test is divided into two parts that are carried out consecutively. The first part consists of three theoretical questions; the first one requires writing a definition or the statement of a theorem, without proof; the second one is similar to the first but, in addition requires a proof; the third one usually requires to prove or disprove an assertion not seen in class. The topics of the first two questions are in a list communicated to the students at the end of the course. In the second part four exercises are assigned, similar to those assigned in class, which cover all the topics seen in the program.
The oral exam consists in the examination and discussion of the written test and, if the commission considers it necessary or the student asks for it, in further questions on the whole course program. 
Assessment criteria:

In the exercises of the first part of the written exam, the skill and organization are evaluated in presenting abstract mathematical concepts with appropriate terms.
In the second part, the ability to apply the aforementioned concepts in exercises is evaluated.
In case the oral examination does not limit itself to the discussion of the written exam, the capacity and the reactivity in dealing with questions of various kinds related to the topics presented is also evaluated. 
Course unit contents:

Algebraic structures. Generalities on matrices. Complex numbers. Trigonometric form of complex numbers. Polynomials with real coefficients. Vector spaces. Subspaces. Linear dependence. Theorem of the exchange. Based and dimension. Linear maps. Correspondence between linear maps and matrices. Change of bases. The theorems on linear maps. Theory of linear systems of equations. Transformation into echelon form. Determinant. Applications of the determinant. Diagonalizability of endomorphisms. Diagonalizability theorem. Diagonalizability of matrices. Affine geometry. Parallelism between linear varieties, pencils of lines and planes. Scalar products: generalities, examples, properties, CauchySchwarz formula. Orthogonality: orthogonal bases, coordinates with respect to orthonormal bases, GramSchmidt procedure, orthogonal projections. Cartesian reference changes, distance in Euclidean space. Real symmetric matrices. 
Planned learning activities and teaching methods:

The educational activities provide lectures at the blackboard where concepts, methods, exercises and their solutions are presented. During the break in each lesson the students are invited to ask questions for clarification on the doubts that may have arisen in the presentation by the teacher.
An additional learning activity is the vision of the written exam with the relative clarifications on the possible difficulties encountered by the candidate. 
Additional notes about suggested reading:

All the teaching material presented during the lessons will be made available on the moodle platform.
The study material includes:
 lecture notes in pdf format,
 list of topics for the first two questions of the first part of the written test,
 complete archive of the exams assigned previously. 
Textbooks (and optional supplementary readings) 

Corrado Zanella, Fondamenti di Algebra Lineare e Geometria. Bologna: Esculapio, 2010.

Innovative teaching methods: Teaching and learning strategies
 Lecturing
 Problem based learning
 Loading of files and pages (web pages, Moodle, ...)
Innovative teaching methods: Software or applications used
 Moodle (files, quizzes, workshops, ...)

