GEOMETRY
cod. 13102

Academic year 2008/09
1° year of course - First semester
Professor
Academic discipline
Geometria (MAT/03)
Field
Matematica, informatica e statistica
Type of training activity
Basic
81 hours
of face-to-face activities
9 credits
hub:
course unit
in - - -

Learning objectives

Supply the student with tools for:<br />a) solve systems of linear equations;<br />b) diagonalize (symmetric) matrices;<br />c) solve easy problems of analytic geometry;<br />d) recognize the type of a conic and write its canonical form.

Prerequisites

Precourse.

Course unit content

<br />1. Real and complex vector spaces. Linear subspaces: sum andintersection. Linear combinations of vectors: lineardipendence/indipendence. Generators, bases and dimension of a vectorspaces. Grassmann formula for subspaces.<br /><br />2. Determinants: Laplace expansion and basic properties. Binet theorem.Row and column elementary operations on matrices. Computation of theinverse matrix. Rank of a matrix.<br /><br />3. Linear systems: Gauss-Jordan method and Rouché Capelli theorem.<br /><br />4. Linear maps. Definition of kernel and image; fundamental theorem onlinear maps. Matrix representation of a linear map and change of bases.Isomorphisms and inverse matrix.<br /><br />5. Endomorphisms of a vector space: eigenvalues, eigenvector andeigenspaces. Characteristic polynomial. Algebraic and geometricmultiplicity. Diagonalizable endomorphisms.<br /><br />6. Scalar products. Orthogonal complement of a linear subspace.Gram-Schmidt orthogonalization process. Representation of isometries byorthogonal matrices. The orthogonal group. Diagonalization of symmetricmatrices: spectral theorem. Positivity criterion for scalar product:Hurewicz theorem.<br /><br />7. Two and three dimensional analytic geometry. Parametric andCartesian equations of a line. Mutual position between two lines in thespace; skew lines. Equation of a plane. Canonical scalar product anddistance. Vector product and its fundamental properties. Distance of apoint from a line and from a plane.<br /><br />8. Conics: elementary properties. Affine and Euclidean classifications.Affine invariants and canonical form of a conic. Center of symmetry andaxes. <br />

Full programme

- - -

Bibliography

F. Capocasa, C.Medori: " Algebra Lineare e Geometria Analitica ", Mattioli<br />A. Nannicini: " Esercizi svolti di algebra lineare, vol.1 ", Pitagora<br />Examination texts are available at the photocopy-office.

Teaching methods

<br />

Assessment methods and criteria

- - -

Other information

- - -