LINEAR ALGEBRA AND GEOMETRY - channel 2

The teaching deals of the following subjects

1) Linear Algebra: resolution of linear systems, study of linear maps, research of their eigenvalues and eigenvectors;

2) Geometry: linear geometry in the planes and in the spaces (lines and planes), study of conics and quadrics.

Algebra

1. Sets. Algebrical binary operation and connected structures: groups, rings and fields. Applications and their properties.

2. Matrices. Determinants and their properties. Inverse of square matrix. Reduces matrices and reduction method. Matrix product. Linear systems. Resolution of linear systems by reduction method, free unknowns.

3. Vector spaces and their properties. Subspaces, intersection, union and sum of subspaces. Linear independence of vectors. Generators and base of a space. The method of completion to a base. Dimension of a vector space. Direct sums.

4. Linear maps and their properties. Injectivity, surjectivity, byectivity, isomorphisms. Study of linear applications. lineari e loro proprietà. Base change and its intrinsic geometrical meaning. Matrices for base change Similar matrices.

5. Eigenvalues, eigenvectors of an endomorphism. Characteristic polynomial. Eingenspaces and their dimension. Independence of the eigenvectors. Simple endomorphisms and their diagonalization matrices.

Geometria:

1) Vector quantities. Vector spaces with two, three and 'n' dimensions. Internal and external operations onto vectors. Scalar and vectorial product, mixed product. Canonic base in the plane and in the space, Components of the vectors and operations by components. Linear geometry in the space. Homogenous and non homogenous coordinates, improper points. Lines in the plane and in the spaces and their equations. The plans and their equations. Angle properties of lines and plane and their respectively distances.

2) Changes of coordinates in the plane: rotations and translations. Conics and their associated matrices, orthogonal invariants. Reduced equations. Bundles of conic. Conics for five points. Circles and cyclic points. Hyperbolas equilatere and their characterisation. Tangent lines for a conic. Improper points in a conic. Classification of conics respect its improper points.

3) Quadrics in space and associated matrices. Irreducible quadrics. Vertices. Degenerate quadrics: broken planes, cones and cylinders. Classification of quadrics not degenerate. Reduced equations of a quadric and reduction of a quadric in canonical form. Tangent lines and plane.

Paxia: Lezioni di Geometria. Spazio Libri, Catania

Giuffrida, Ragusa: Corso di Algebra Lineare. Il Cigno Galileo Galilei, Roma

Bonacini, Cinquegrani, Marino: Geometria: esercizi svolti. Ed. Cavallotto