Linear Algebra and Elements of Geometry (2008/2009)

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Teaching is organised as follows:
Unit Credits Academic sector Period Academic staff
modulo di base 6 MAT/02-ALGEBRA 1° Q - solo 1° Anno Francesca Mantese
modulo avanzato 3 MAT/03-GEOMETRY 2° Q Mauro Spera

Learning outcomes

Module: modulo di base
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The aim is to introduce the basic facts about Linear Algebra and its applications


Module: modulo avanzato
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The course presents an introduction to plane and space analytic geometry, in a projective, affine and metric environment, respectively. In particular, conics and quadrics are discussed in the different frameworks.
Both analytical (coordinates, matrices) and synthetic tools are employed. The ultimate aim is strengthening the student's
geometric intuition, abstraction, and computational expertise, in view of future developments and applications,
in different contexts.

Syllabus

Module: modulo di base
-------
* Matrices and linear systems: matrices, matrix operations, linear systems of equations, Gauss elimination, inverse matrices, LU decomposition.
* Vector spaces: definition and examples, subspaces, sets of generators. Linear dependence and independence, bases, dimension.
* Linear maps and associated matrices: composition of linear maps and matrix multiplication, base change, kernel and image of a linear map, rank of matrices, dimension formula.
* Inner products and orthogonality: inner product between vectors, orthogonal and orthonormal bases, orthogonal projections, Gram-Schmidt algorithm.
* Canonical forms: eigenvalues and eigenvectors, characteristic polynomial, geometric and algebraic multiplicity of eigenvalues, diagonalizability criteria.


Module: modulo avanzato
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LINEAR ALGEBRA AND ELEMENTS OF GEOMETRY (2nd module)
The course presents an introduction to plane and space analytic geometry, in a projective, affine and metric environment, respectively. In particular, conics and quadrics are discussed in the different frameworks.
Both analytical (coordinates, matrices) and synthetic tools are employed. The ultimate aim is strengthening the student's
geometric intuition, abstraction, and computational expertise, in view of future developments and applications,
in different contexts.
Review of geometric vectors, scalar and vector product. Affine spaces. Affine subspaces. Affine notions: incidence,
parallelism. Geometric interpretation of linear systems. Ordinary line, plane, space. Plane and space lines.
Pencils of lines and planes. Incidence, parallelism, coplanarity.
Barycentric coodinates.
Affine euclidean spaces. Distance between affine subspaces; examples.
The common perpendicular to skew lines. (Convex) angle between two lines, planes, a line and a plane.
The projective space associated to a finite dimensional vector space. Homogeneous coordinates. Projective
embedding (completion) of an affine space. The ordinary projective line, plane, space. Lines in projective plane.
Conics (elementary theory). Conics in projective plane and their classification. Tangent to a conic.
Polarity. Reciprocity theorem. geometric construction of polar lines. Self-polar triangles and geometric interpretation
of Sylvester's theorem. Pencils of conics. Affine classification of conics (deduced from incidence relations with
the line at infinity): ellipses, hyperbolas, parabolas. centre, diameters; conjugate diameters. Asymptotes.
Metric classification of conic; axes. Orthogonal invariants. Circles. Isotropic lines. Circular points. Foci. Directrices.
Comparison with the classical approach. Conics as Be'zier curves.
Quadrics and their projective, affine and metric classification.
Matrix approach to plane and spatial homographies.
Further topics in linear algebra: Sylverster's inertia theorem and the spectral theorem.
NOTES: 1. Lecture notes are available on the course web page.
2. The program is tentative and subject to change.



References


M.SPERA Lecture Notes (handwritten)

M.C.BELTRAMETTI, E.CARLETTI, D.GALLARATI, F.MONTI BRAGADIN,
Lezioni di geometria analitica e proiettiva, Bollati-Boringhieri, Torino, 2002.

M.R.CASALI, C.GAGLIARDI, L.GRASSELLI, Geometria,
Progetto Leonardo, Esculapio, Bologna, 2002.

R.CASSE, Projective Geometry, an introduction Oxford University Press,
Oxford, 2006

G.CASTELNUOVO, Lezioni di Geometria Analitica , Soc. Ed. Dante Alighieri, Milano, Roma, 1969.

M.DOCCI, R.MIGLIARI, La Scienza della rappresentazione.
Fondamenti e applicazioni della geometria descrittiva, Carocci, Roma, 1999.

F.ENRIQUES, Lezioni di Geometria Proiettiva, Zanichelli, Bologna, 1996.

J.GALLIER, Geometric Methods and Applications for Computer Science and
Engineering, Springer, Berlin, 2000.

E.GREGORIO, L.SALCE , Algebra Lineare Ed.Libreria Progetto, Padova, 2005

R.HARTLEY, A.ZISSERMAN, Multiple View Geometry in Computer Vision,
Cambridge, Cambridge, 2003.

D.HILBERT, S.COHN-VOSSEN Geometria intuitiva, Boringhieri, Torino, 1972.

D.MARSH, Applied Geometry for Computer Graphics and CAD,
Springer, London, 2005.

E.SERNESI, Geometria 1,2 Bollati Boringhieri, Torino, 1989, 1994.

Assessment methods and criteria

Module: modulo di base
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Written test


Module: modulo avanzato
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Assessment:Written exam at the end of the course, followed by an oral test (to be arranged with Dr. F. Mantese, instructor of the basic module).

Reference books
Author Title Publisher Year ISBN Note
E.Gregorio, L.Salce Algebra Lineare Libreria Progetto Padova 2005