
Course unit
GEOMETRY (Iniziali cognome AL)
SCN1032568, 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 
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 
Prerequisites:

None 
Target skills and knowledge:

Knowledge of the basic notions and results on vector spaces. Familiarity with matrix calculus. Knowledge of the interaction between linear algebra and geometry. 
Examination methods:

Written exam consisting both of exercises and theoretical questions. Students with a grade of 28 or higher have also to take an oral exam.
It will be possible to replace the written exam by two small tests, one halway through the course and one at the end of the course. 
Assessment criteria:

Knowledge of the main definitions and theorems.
Capacity to solve exercises in which one applies linear algebra.
Capacity to show results concerning vector spaces. 
Course unit contents:

Solving a system of linear equations. Gauss elimination method.
Matrix calculus, invertible matrices. Rank of a matrix.
Vector spaces, subspaces, linear dependence, bases. Dimension of a vector space.
Sums of vector spaces, intersection of vector spaces.
Linear maps. Kernel and image of a linear map. Matrix of a linear map. Matrix associated with a change of basis. Determinant of a matrix. Eigenvalues and eigenvectors of a linear map. Diagonalizable matrices.
The space of geometric vectors: inner product and its properties, the norm of a vector, Schwarz inequality.
Quadratic forms. Symmetric bilinear forms. Spectral theorem for real symmetric matrices. Affine spaces and subvarieties. Affine coordinates. Affine transformations. Euclidean space. Isometries. Parallel, incident and skew subvarieties. Distance, angles. Volume of parallelepipeds: explicit formulas. Classification of conics. 
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:

The instructor will provide notes, which will be available on the moodle page of the course. 
Textbooks (and optional supplementary readings) 

Candilera, Maurizio; Bertapelle, Alessandra, Algebra lineare e primi elementi di geometriaMaurizio Candilera, Alessandra Bertapelle. Milano: McGrawHill, ©2011, .

Mauri, Luca; Schlesinger, Enrico, Esercizi di algebra lineare e geometria. Bologna: Zanichelli, 2013.

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

