# Matematica | GEOMETRY II

## Matematica GEOMETRY II

 0512300040 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2020/2021

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2018 SECONDO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/03 8 64 LESSONS BASIC COMPULSORY SUBJECTS

 LUCA VITAGLIANO T
Objectives
THE AIM OF THIS COURSE IS TO INTRODUCE THE STUDENTS TO THE THEORY OF EUCLIDEAN VECTOR SPACES AND TO THE THEORY OF AFFINE AND EUCLIDEAN GEOMETRY.

EXPECTED LEARNING RESULTS

KNOWLEDGE AND UNDERSTANDING:

THE COURSE AIMS TO PROVIDE THE STUDENTS WITH A SOLID BASIC KNOWLEDGE OF EUCLIDEAN VECTOR SPACES, AFFINE SPACES AND AFFINE AND ISOMETRIC MAPS.

APPLYING KNOWLEDGE AND UNDERSTANDING:

THE COURSE FURTHER AIMS TO ENABLE STUDENTS TO SOLVE PROBLEMS CONCERNING EUCLIDEAN VECTOR SPACES AND AFFINE SPACES, PARTICULARLY IN DIMENSION 2 AND 3.
Prerequisites
IT IS REQUIRED A KNOWLEDGE OF THE TOPICS COVERED IN THE COURSE GEOMETRY I
Contents
1. EUCLIDEAN VECTOR SPACES

SCALAR PRODUCTS. THE NORM AND ITS PROPERTIES. ANGLE BETWEEN TWO VECTORS. CAUCHY-SCHWARZ INEQUALITY. ORTHOGONALITY. GRAM-SCHMIDT ORTHOGONALIZATION. COMPONENTS OF A VECTOR IN AN ORTHONORMAL BASIS. ORTHONOGONAL MATRICES AND ORTHONORMALITY OF NUMERICAL VECTORS. CHANGE OF ORTHONORMAL FRAMES. ORTHOGONAL SUBSPACES. ORTHOGONAL COMPLEMENT.
ORTHOGONAL MAPS. THE ORTHOGONAL GROUP. THE SPECIAL ORTHOGONAL GROUP. CLASSIFICATION OF ORTHOGONAL TRANSFORMATIONS IN DIMENSION 2 AND 3.

2. THE PROBLEM OF DIAGONALIZATION

DIAGONALIZATION OF AN ENDOMORPHISM. EIGENVALUES, EIGENVECTORS, EIGENSPACES. HOW TO COMPUTE THE EIGENVALUES, CHARACTERISTIC POLYNOMIAL, ALGEBRAIC AND GEOMETRIC MULTIPLICITY. DIAGONALIZABILITY CRITERIA. DIAGONALIZATION OF ENDOMORPHISMS OF A EUCLIDEAN SPACE. ORTHOGONAL DIAGONALIZABILITY. SYMMETRIC ENDOMORPHISMS, SYMMETRIC MATRICES.
EIGENVALUES OF A SYMMETRIC ENDOMORPHISM. ORTHOGONAL ENDOMORPHISMS AND THEIR REPRESENTATION. ORTHOGONAL MATRICES. EIGENVALUES OF AN ORTHOGONAL ENDOMORPHISM. ORTHOGONALLY DIAGONALIZABLE ENDOMORPHISMS. THE SPECTRAL THEOREM.

3. HERMITIAN FORMS

HERMITEAN FORMS AND REPRESENTATIONS. HERMITEAN MATRICES. HERMITEAN PRODUCTS. THE STANDARD HERMITEAN PRODUCT. HERMITEAN SPACES. UNITARY AND HERMITEAN ENDOMORPHISMS

4. AFFINE AND EUCLIDEAN AFFINE SPACES

AFFINE SPACES. AFFINE SUBSPACES. AFFINE FRAMES. REPRESENTATIONS OF SUBSPACES. PARALLELISM AND INTERSECTION OF SUBSPACES. GEOMETRY IN AN AFFINE SPACE OF DIMENSION 2 AND 3. EUCLIDEAN AFFINE SPACES, CARTESIAN FRAMES, DISTANCE BETWEEN POINTS, ANGLE BETWEEN LINES. GEOMETRY IN A EUCLIDEAN AFFINE SPACE OF DIMENSION 2 AND 3. AFFINE MAPS. AFFINE EXTENSION THEOREM. THE AFFINE GROUP. TRANSLATIONS AND STABILIZERS OF A POINT. EVERY AFFINITY IS A COMPOSITION OF A TRANSLATION AND A CENTROAFFINITY. ISOMETRIES. DIRECT AND INVERSE ISOMETRIES. CLASSIFICATION OF THE ISOMETRIES OF A PLANE.
Teaching Methods
THE COURSE CONSISTS IN 64 FRONT TEACHING HOURS, DIVIDED INTO 50 HOURS OF THEORETICAL LECTURES AND 14 HOURS OF EXERCISES. DURING THE LECTURES WE WILL DISCUSS THEORETIC TOPICS, TOGETHER WITH CONCRETE PROBLEMS TO BE SOLVED BY THE STUDENTS UNDER THE INSTRUCTOR GUIDANCE, DURING THE EXERCISE SESSIONS.
Verification of learning
THE EXAM IS AIMED TO EVALUATE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED IN CLASS AND THE ABILITY TO APPLY SUCH KNOWLEDGE TO THE SOLUTION OF SIMPLE PROBLEMS.

THE EXAMINATION IS DIVIDED INTO A SELECTIVE WRITTEN EXAM AND AN ORAL EXAM. THE WRITTEN EXAM CONSISTS OF SOME EXERCISES. THE ORAL EXAM EVALUATES THE ACQUIRED KNOWLEDGE OF THE THEORY OF EUCLIDEAN VECTOR SPACES AND THE THEORY OF AFFINE SPACES.

THE FINAL ASSESSMENT IS EXPRESSED BY A VOTE FROM 0 TO 30. THE WRITTEN EXAM IF PASSED GIVES ACCESS TO THE ORAL EXAM WHICH DETERMINES THE FINAL VOTE IN FULL.
Texts
R. ESPOSITO, A. RUSSO, LEZIONI DI GEOMETRIA, PARTE PRIMA, LIGUORI.

E. SERNESI, GEOMETRIA 1, BOLLATI BORINGHIERI.

S. LIPSCHUTZ, ALGEBRA LINEARE MCGRAW-HILL.
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