B005475 - GEOMETRY I

Italian

The space R^n.
Vector spaces.
Matrices.
Linear Applications.
Linear Applications and matrices.
Scalar products and orthogonality.
Matrices and bilinear applications.
Polynomials and matrices.
Triangulation for matrices and linear applications.
Spectral theorem.
Jordan canonical form.
Projective geometry.
Affine geometry.
Euclidean geometry.
Conics and quadrics.

Serge Lang, ALGEBRA LINEARE, Bollati Boringhieri

Gareth Williams, LINEAR ALGEBRA with applications, Jones and Bartlett Mathematics

Nigel Hitchin, Projective Geometry (online notes)

The course aims to provide the knowledge and the basic technical skills in Linear Algebra and Analytic and Projective Geometry. The topics covered in the course and the technical skills provided are needed, or strongly suggested, for the pursuance of the studies.

None

Lectures, training sessions.

Intermediate written tests. Final written and oral exam.

The space R^n. Points of the n-space; vectors; canonical scalar product; norm of a vector; orthogoality and parallelism; distance; Euclidean Geometry of the n-space; lines, planes, iper-planes.

Vector spaces. Subsets of linearly dependent, or linearly independent, vectors; bases of a vector space; dimension of a vector spacesums of vector spaces; direct sums of vector spaces.


Matrices. The vector space of matrices; sum and product of matrices; linear equations; systems of linear equations and matrices; the Gauss' algorithm to solve systems of linear equations.

Linear maps. Kernel and image of a linear map; dimension of the kernel and of the image; composition of linear maps.

Linear maps and matrices. Linear map associated to a matrix; matrix associated to a linear map; composition of linear maps and matrices.

Determinants. Determinants of order 2; Properties of determinants; The Cramer rule; existence of determinants; unicity of the determinant; determinat of the transpose of a matrix; determinant of the product of two matrices; determinant of a linear map.

Scalar products and orthogonality. Scalar products; positive definite scalar products; orthogonal bases in the general case; dual space; rank of a matrix and systems of linear equations.

Matrices and bilinear maps. Bilinear forms; quadratic forms; symmetric operators; Hermitian operators; The Sylvester's Theorem.

Polynomials and matrices. Polynomials of matrices and linear maps; eigenvectors and eigenvalues; characteristic polynomial.

Triangulation of matrices and linear maps. Fans; basis Fans; existence of triangulation; Hamilton-Cayley's Theorem; diagonalization of unitary matrices.

The Spectral theorem. Eigenvectors of linear symmetric maps; The Spectral Theorem; the complex case.

The Jordan normal form. Generalized eigenspaces of a linear map; Jordan normal form for a nilpotent map; Jordan basis and Jordan form for a linear map.

Projective geometry. Projective spaces; projective subspaces; the group of projective transformations; points in general position; Desargues' and Pappus Theorems; duality.

Quadrics. Quadratic forms; conics and quadrics; polarity with respect to a conic; projective classification of conics; pencils of conics.
Affine geometry. The affine plane and the affine space; the group of affine transformations; affine classification of conics.

Euclidean geometry. The group of isometries; euclidean classification of conics.

Exercises and examples.