Academic Year 2016/2017 - 1° Year

MAT/03 - 9 CFU - 2° Semester

CONTENTS OF THE COURSE

Algebra

1. General information on sets, operations. Applications between sets, image and inverse image, injectivity, surjectivity, biettive applications. Sets with operations, the main geometric structures: groups, rings and fields.

2. General information on matrices. Rank. Determinants and their properties. The Laplace * theorems. Inverse of a square matrix. calculating the inverse of a square matrix. Binet theorem *. reduced matrices and reduction method. Product of matrices. linear systems, Rouché-Capelli theorem. Solving linear systems with the reduction method (Gauss), free unknowns. homogeneous systems. Cramer's theorem.

3. Vector spaces and their properties. Examples. Subspaces. Intersection, union and sum of subspaces. linear independence, its criterion. Generators of a space. Base of a space, the method of successive waste, completion to a base. Lemma Steinitz *, dimension of a vector space. Grassmann * formula. Direct sums. Theorem of Kronecker. Proof of Theorem of Rocuhé-Capelli I

4. Linear maps and their properties. Core and image of a linear. Injectivity, surjectivity, isomorphisms. Study of linear applications.

5. Eigenvalues, eigenvectors and eigenspaces of an endomorphism. characteristic polynomial. Independence of the eigenvectors.

Geometry:

1. The vectors of ordinary space. Components of the vectors and using components operations. linear geometry in space. Cartesian coordinates and homogeneous coordinates. The plans and their equations. The straight, their representation. improper elements. of lines and planes angle properties. Distances. Linear geometry in the plane. Cartesian coordinates and homogeneous coordinates.

2. Changes in the coordinate plane, rotations and translations. Conics and associated matrices, orthogonal invariants. reduced equations, reduction of a conic to canonical form. Classification of irreducible conics. Tangent lines. Bundles of conics and their use to determine special conics.

3. Quadrics in space and associated matrices. Irreducible quadrics. Vertices and degenerate quadrics. Cones and cylinders, their sections. reduced equations, reduction of a quadric in canonical form. Classification of quadrics does not degenerate. Sections of quadrics with lines and planes. Straight lines and tangent planes.

Proofs of theorems marked with * may be omitted