Portale di Ateneo - En.Unibs.it

Other Courses of the Study Programme

Course year: 1

ALGEBRA AND GEOMETRY - 9 credits - Basic
BASIC CHEMISTRY - 6 credits - Basic
ELEMENTS OF COMPUTING - 6 credits - Basic
EXPERIMENTAL PHYSICS 1 (MECH.,ELECTROMAG.) - 9 credits - Basic
INTRODUCTION TO BASIC MATTERS - MATHEMATICS - Student's own choice
MATHEMATICAL ANALYSIS
Attendance grouping
12 credits - Raggruppamento di frequenza
METALLURGY - 6 credits - Related/Supplementary
PROBABILITY AND STATISTICS - 6 credits - Basic
TECHNICAL DRAWING - 6 credits - Related/Supplementary
TEST OF ENGLISH - 3 credits - Language/Final Examination
TEST OF FRENCH - 3 credits - Language/Final Examination
TEST OF GERMAN - 3 credits - Language/Final Examination
TEST OF SPANISH - 3 credits - Language/Final Examination

Course year: 2

ELECTRICAL AND ELECTRONIC MEASUREMENTS - 6 credits - Related/Supplementary
FOUNDATIONS OF ECONOMICS - 9 credits - Related/Supplementary
FUNDAMENTALS OF MECHANICS AND MACHINE DESIGN
Attendance grouping
12 credits - Raggruppamento di frequenza
INTRODUCTION TO THERMODYNAMICS AND HEAT TRANSFER - 9 credits - Characterising
MECHANICAL TECHNOLOGY - 9 credits - Characterising
SYSTEMS ANALYSIS AND OPTIMIZATION
Attendance grouping
12 credits - Raggruppamento di frequenza

Course year: 3

APPLIED ACOUSTICS - 6 credits - Student's own choice
BUSINESS ANALYSIS AND MODELING - 6 credits - Student's own choice
BUSINESS ECONOMICS AND ORGANIZATION - 6 credits - Characterising
BUSINESS ENGLISH - 6 credits - Student's own choice
DECISION ANALYSIS FOR BUSINESS - 6 credits - Student's own choice
DRAWING AND CAD LABORATORY - 9 credits - Student's own choice
ELEMENTS OF BIOMECHANICS - 3 credits - Student's own choice
ELEMENTS OF METALLURGICAL TECHNOLOGY - 6 credits - Student's own choice
ENVIRONMENTAL ECONOMICS - 6 credits - Student's own choice
ENVIRONMENTAL QUALITY MODELING - 6 credits - Student's own choice
FINAL THESIS - 3 credits - Language/Final Examination
FINANCIAL STATEMENT ANALYSIS AND FINANCIAL COMMUNICATION - 9 credits - Student's own choice
HISTORY OF ECONOMIC THOUGHT - 9 credits - Student's own choice
INDUSTRIAL PLANTS DESIGN AND PRODUCTIVE SYSTEMS LABORATORY
Attendance grouping
12 credits - Raggruppamento di frequenza
INDUSTRIAL POLYMERS - 6 credits - Student's own choice
INTRODUCTION TO DATABASES - 6 credits - Student's own choice
INTRODUCTION TO FINANCIAL MANAGEMENT - 6 credits - Student's own choice
LABORATORY OF SPECTROSCOPY FOR MATERIALS ANALYSIS - 6 credits - Student's own choice
LABOUR LAW - 9 credits - Student's own choice
MANAGEMENT CONTROL SYSTEMS AND FINANCE - 9 credits - Characterising
MARKETING - 9 credits - Student's own choice
MECHANICAL AND THERMAL MEASUREMENTS - 9 credits - Student's own choice
PLANTS AND INDUSTRIAL LOGISTICS MANAGEMENT
Attendance grouping
12 credits - Raggruppamento di frequenza
PRIVATE LAW - 9 credits - Student's own choice
PROBLEM SOLVING FOR COMPUTER SCIENCE - 6 credits - Student's own choice
PUBLIC ECONOMICS - 9 credits - Student's own choice
PUBLIC LAW - 9 credits - Student's own choice
QUALITY INDUSTRIAL MANAGEMENT - 6 credits - Characterising
SCIENTIFIC COMPUTING AND LABORATORY - 6 credits - Student's own choice
SOCIOLOGY OF ORGANIZATION - 6 credits - Student's own choice
TRAINING ACTIVITY/INTERNSHIP 12 CREDITS - UNDERGRADUATE LEVEL - 12 credits - Student's own choice
TRAINING ACTIVITY/INTERNSHIP 3 CREDITS - UNDERGRADUATE LEVEL - 3 credits - Student's own choice
TRAINING ACTIVITY/INTERNSHIP 6 CREDITS - UNDERGRADUATE LEVEL - 6 credits - Student's own choice
TRAINING ACTIVITY/INTERNSHIP 9 CREDITS - UNDERGRADUATE LEVEL - 9 credits - Student's own choice

ALGEBRA AND GEOMETRY

Single discipline educational activity

Course Sheet

Academic Year of enrolment:

2015/2016

Academic Year of provision:

2015/2016

Type of Course:

Basic

Degree:

Bachelor's Degree Programme in MANAGEMENT ENGINEERING

Disciplinary Sector:

Algebra

Language:

Italian

Grouping type:

Credits:

Programme Year:

Professor and Collaborators:

PELLEGRINI Silvia Collaborator

PASOTTI Anita Professor

GAMMACURTA COSTANTINO Collaborator

Cycle:

First semester

Hours of classroom activity:

108

Hours of individual study:

117

Good knowledge of basic math concepts learned in higher education.

Educational Goals

This course is dedicated to first year undergraduate students in Engineering. It introduces the main tools of linear algebra: vector spaces, linear systems, diagonalizability, bilinear and quadratic forms. All of these are essential instruments for any further course. The second part of the course is dedicated to analytic geometry in affine and Euclidean spaces of dimension 2 or 3: spheres, revolution surfaces, conics and quadrics are discussed. These are also classical topics, necessary for any scientific or technical degree.

Content

Linear algebra and its applications. In particular, in the first part of the course we shall introduce the theory of vector spaces, matrices and linear systems. Then, we shall apply these results to the study of analytical geometry. More in detail we shall study affine and euclidean geometry in dimension 2 and 3 (in a complexified projective completion) and describe conics and quadrics in these spaces.

FULL PROGRAMME

1. Naive Set theory
Functions and relations; basic algebraic structures

2. Vector spaces
Definition; linear dependence; generator sets and bases; Steinitz' lemma; dimension.
Subspaces; intersection and sum; direct sum; Grassmann's formula.

3. Matrices
Matrix vector spaces; row/column product; rank and determinant.
Linear systems; eigenvalues and eigenvectors; diagonalizable matrices.

4. Bilinear and quadratic forms
Real symmetric matrices; orthogonal and orthonormal bases; Gram-Schmidt's ortonormalization.
The spectral theorem.

5. Analytic geometry in dimension 2 and 3

Affine and euclidean spaces. Analytic geometry in the plane and in 3-space: lines, planes, circles, spheres and revolution surfaces. Completion and complexification of an Euclidean space. Conics: polarities, centre, axes and asymptotes; reduction of the equation of a conic to canonical form. Quadrics: classification depending on multiple points; investigation of planar sections; identification of an irreducible quadric.

Extended Syllabus

ALGEBRA
Set theory: correspondences and functions. Relations on a set. Definition of the main algebraic structures.

Vector spaces: definition of a vector space. Linear dependence and independence. Generators. Finitely generated vector spaces: Steinitz Lemma, bases and dimension of a vector space. Subspaces of a vector space. Intersection, sum and direct sum of subspaces. Grassmann formula.

Matrices and linear systems: the vector space of matrices. Product, rank and determinant. Linear systems and their risolubility: Theorems of Rouche-Capelli and of Cramer. Eigenvalues and eigenvectors of a matrix. Diagonalizability and criteria.
Bilinear and quadratic forms: real and symmetric matrices. Orthogonal and orthonormal bases. Process of Gram-Schmidt orthogonalization. Orthogonal matrices. Orthogonally diagonalizable matrices.

GEOMETRY
Affine spaces: definitions, translations, subspaces, parallelism. Coordinatization of an affine space and analytic geometry in the plane and in three-dimensional space.
Euclidean spaces: distance, ortonogality, circles, spheres, surfaces of revolution and fundamental loci.

Projective spaces: projective extension of an affine geometry, projective subspaces, homogeneous coordinates and representation in homogeneous coordinates of the subspaces. Complexification.

Real algebraic curves and surfaces: an order of a curve, simple and singular points. Conic, projective classification, polarity, affine classification, canonical forms.
Quadrics: affine classification, cones and cylinders, flat sections.

Recommended Bibliography

Algebra Lineare e Geometria Analitica- Ed. Apollonio (2014) - S. Pellegrini
Esercizi di Algebra Lineare - Ed. Apollonio - S. Pellegrini
Esercizi di Geometria Analitica - Ed. Apollonio - S. Pellegrini, A. Benini

Methods of Provision

Conventional

Teaching Methods

Lectures of theory and exercises

Assessment Methods

The exam consists in a written test, then, if it has a positive result, there is, in the same session, an oral examination.

Assessment:

Final Mark

Programme of Educational Activities

Start date of teaching period:

Wednesday, September 16, 2015

End date of teaching period:

Tuesday, December 22, 2015

Other Information

More information about the course can be found on the website: http://dm.ing.unibs.it/~anita.pasotti/teaching.html

Amministrazione trasparente

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Other Courses of the Study Programme

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Course year: 2

Course year: 3

ALGEBRA AND GEOMETRY

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