Geometry 1 - (9 cfu)

 Prof. Leonardo Biliotti - E-mail. leonardo.biliotti@unipr.it

Finalità

Supply the student with tools for:
a) solve systems of linear equations;
b) diagonalize (symmetric) matrices;
c) solve easy problems of analytic geometry;
d) Operations on vectors and matrices.

Programma

1. Real and complex vector spaces. Linear subspaces: sum and intersection. Linear combinations of vectors: linear dipendence/indipendence. Generators, bases and dimension of a vector spaces. Grassmann formula for subspaces.

2. Determinants: Laplace expansion and basic properties. Binet theorem. Row and column elementary operations on matrices. Computation of the inverse matrix. Rank of a matrix.

3. Linear systems: Gauss-Jordan method and Rouché Capelli theorem.

4. Linear maps. Definition of kernel and image; fundamental theorem on linear maps. Matrix representation of a linear map and change of bases. Isomorphisms and inverse matrix.

5. Endomorphisms of a vector space: eigenvalues, eigenvector and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity. Diagonalizable endomorphisms.

6. Scalar products. Orthogonal complement of a linear subspace. Gram-Schmidt orthogonalization process. Representation of isometries by orthogonal matrices. The orthogonal group. Diagonalization of symmetric matrices: spectral theorem. Positivity criterion for scalar product. A brief discussion on the complex case..

7. Three dimensional analytic geometry. Parametric and Cartesian equations of a line. Mutual position between two lines in the space; skew lines. Equation of a plane. Canonical scalar product and distance. Vector product and its fundamental properties. Distance of a point from a line and from a plane.

8. Topics in algebra and/or geometry.

Modalità d'esame

Usually the evaluation consists of a written exam.

Propedeuticità

Precourse.

Testi consigliati

F. Capocasa, C.Medori: " Algebra Lineare e Geometria Analitica ",
A. Alessandrini, L. Nicolodi ''Geometria A''
A. Alessandrini ''Geometria B''
A. Abate, C. De Fabritiis ''Geometria analitica con elementi di algebra lineare''

Ultimo aggiornamento: 05-10-2009

Chiudi la finestra