
Course unit
GEOMETRY (Iniziali cognome AL)
SCN1032568, A.A. 2016/17
Information concerning the students who enrolled in A.Y. 2016/17
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Basic courses 
MAT/03 
Geometry 
8.0 
Course unit organization
Period 
First semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
2.0 
16 
34.0 
No turn 
Lecture 
6.0 
48 
102.0 
No turn 
Examination board
Board 
From 
To 
Members of the board 
11 Geometria (iniziali cognomi MZ) 
01/10/2018 
30/11/2019 
URBINATI
STEFANO
(Presidente)
KLOOSTERMAN
REMKE NANNE
(Membro Effettivo)
BALDASSARRI
FRANCESCO
(Supplente)
GARUTI
MARCOANDREA
(Supplente)

10 Geometria (iniziali cognome AL)) 
01/10/2018 
30/11/2019 
KLOOSTERMAN
REMKE NANNE
(Presidente)
URBINATI
STEFANO
(Membro Effettivo)
BALDASSARRI
FRANCESCO
(Supplente)
GARUTI
MARCOANDREA
(Supplente)

9 Geometria (iniziali cognome AL)) 
01/10/2017 
30/11/2018 
KLOOSTERMAN
REMKE NANNE
(Presidente)
URBINATI
STEFANO
(Membro Effettivo)
BALDASSARRI
FRANCESCO
(Supplente)
GARUTI
MARCOANDREA
(Supplente)

8 Geometria (iniziali cognomi MZ) 
01/10/2017 
30/11/2018 
URBINATI
STEFANO
(Presidente)
KLOOSTERMAN
REMKE NANNE
(Membro Effettivo)
BALDASSARRI
FRANCESCO
(Supplente)
GARUTI
MARCOANDREA
(Supplente)

7 Geometria (MZ) 
01/10/2016 
30/09/2017 
GARUTI
MARCOANDREA
(Presidente)
BALDASSARRI
FRANCESCO
(Membro Effettivo)
BERTAPELLE
ALESSANDRA
(Supplente)
CANDILERA
MAURIZIO
(Supplente)
URBINATI
STEFANO
(Supplente)

Prerequisites:

None 
Target skills and knowledge:

A standard course on Affine and Euclidean Geometry, with the methods of Linear Algebra. As for the latter, it will be essential to digest the notions of vector space, linear map, bilinear form, and to understand the classification of these objects, over both the real and the complex numbers. A for geometrical aspects, the crucial notions will be those of affine space, of affine sub variety (points, lines, planes,...), of scalar product of vectors, of distance between sub varieties, and of volume of bodies. We will study affine maps, orthogonal transformations, coordinate changes. We will shortly discuss conics and their euclidean classification. We will give a short hint of noneuclidean affine spaces (the Minkowski plane). 
Examination methods:

It will be possible to take the exam in two partials, one in November and one midJanuary. The written examination consists in solving a few exercises. The "oral" exam is in fact usually taken in written form, as well. It consists in more theoretical questions: proof of a simple theorem, definitions, short conceptual exercises. The written exam usually contains a couple of questions valid for the oral exam: the student has the choice of either answering them, or pospone the oral test to a later session. A true oral examination (at the blackboard) is reserved for those who have already a high mark in the written test, and aim at an excellence final grade. 
Assessment criteria:

Absolutely essential will be the understanding of definitions and of the statements of theorems, as well as the skill to solve the exercises in the textbook. Proofs of theorems will only be required from those who aim to a high grade and will be asked during the oral examination. 
Course unit contents:

Vector spaces, subspaces, linear dependence, bases. Dimension of a (finitely generated) vector space. The space of geometric vectors : scalar product and its properties, the norm of a vector, Schwarz inequality, vector product and mixed product. Sum and intersection of subspaces. Dual vector space. Linear maps. Projections and symmetries. Invertible matrices and basechange. Rank of a matrix. Solution of a linear system. Gauss elimination method. Alternating multilinear functions. Determinant of a linear map and its properties. Eigenvalues and eigenvectors, characteristic polynomial of an endomorphism. Diagonalizable matrices. Quadratic forms. Symmetric bilinear forms. Spectral theorem for real symmetric matrices. Hints to hermitian forms. Affine spaces and sub varieties. Affine coordinates. Affine transformations. Euclidean space. Isometries. Parallel, incident and skew sub varieties. Distance, angles. Volume of parallelepipeds: explicit formulas. Classification of conics. A hint to the Minkowski plane. 
Planned learning activities and teaching methods:

Theoretical lessons (50% of the time) alternated with sessions of problemsolving (50% of the time). 
Additional notes about suggested reading:

We propose two textbooks.
1) The book by CandileraBertapelle is more advanced and suitable for those who have mathematical interests. It deals with higher dimensions and dedicates attention to Transformation Groups (affine maps, isometries, rigid movements,...). The exercises are often difficult and are in fact compliments to the theory.
2) The book by AbateDe Fabritiis is a completely standard course of Analytic Geometry and Linear Algebra in dimensions 2 and 3, as was initiated in high school. It is the ideal textbook for Engineers. It lacks completely of examples in higher dimensions and of the notion of Transformation Groups. The course will cover the latter topics as well !
In class only the simplest proofs will be detailed. We refer the interested student to the textbook for details and further proofs. WE will give instead lots of examples and exercises. We recommend to work at least through the exercises of the book AbateDe Fabritiis, which are very simple and basic.
Some extra material, and the problems assigned in previous exams, will be available on the Moodle site of the course. 
Textbooks (and optional supplementary readings) 

M. Candilera, A. Bertapelle, Algebra lineare e primi elementi di Geometria. : McGrawHill Com, 2011.

M. Abate, C. De Fabritiis, Geometria Analitica con elementi di algebra lineare. : McGrawHill Com, 2015.


