LINEAR ALGEBRA AND GEOMETRY A - E

MAT/03 - 9 CFU - 2° Semester

Teaching Staff

SANTI DOMENICO SPADARO


Learning Objectives

The course deals with the following topics:

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

2) Geometry: linear geometry in the plane and in 3-dimensional space (lines and planes), study of conics and quadrics.


Course Structure

The course is structured into lectures and recitation sessions.



Detailed Course Content

The detailed course content will be posted on the instructor's website at the following address: http://santispadaro.weebly.com/diario.html

 

Algebra

 

1. Sets. Binary algebraic operations and related structures: groups, rings and fields. Functions and their properties.

2. Matrices. Determinants and their properties. The inverse of a square matrix. Echelon form and the Gaussian reduction mathod. Matrix product. Linear systems. Resolution of linear systems by the 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. Injective, surjective and bijective maps. Isomorphisms. Study of a linear map. Base change and its intrinsic geometrical meaning. The base change matrix. Similar matrices.

5. Eigenvalues and eigenvectors of an endomorphism. The characteristic polynomial. Eingenspaces and their dimension. Simple endomorphisms and diagonalizable matrices.

Geometry

1) Vector quantities. Vector spaces with two, three and 'n' dimensions. Internal and external operations onto vectors. Dot product, cross product and scalar triple product. Canonical bases, components of a vector. Linear geometry in three-dimensional space. Homogenous and non homogenous coordinates, improper points. Lines in the plane and in the space and their equations. Planes and their equations. Angles and distances between lines and planes.

2) Change of coordinates in the plane: rotations and translations. Conics and their associated matrices, orthogonal invariants. Reduced equations. Bundle of conics. Conics through five points. Circles and cyclic points. Equilateral hyperbolae and their characterisation. Tangent lines to a conic. Improper points of a conic. Classification of conics by means of their improper points.

3) Quadrics in space and associated matrices. Irreducible quadrics. Vertices. Degenerate quadrics: broken planes, cones and cylinders. Classification of non-degenerate quadrics. Reduced equations of a quadric and reduction of a quadric to canonical form. Tangent lines and planes. Plane sections. Spheres.



Textbook Information

Paxia: Lezioni di Geometria. Spazio Libri, Catania

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




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