Matematica | GEOMETRY II

cod. 0512300040

GEOMETRY II

	0512300040
	DIPARTIMENTO DI MATEMATICA
	EQF6
	MATHEMATICS
	2022/2023

PERCORSO SHARED STUDY PATHWAY

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2018
	SPRING SEMESTER

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/03	8	64	LESSONS	BASIC COMPULSORY SUBJECTS

EXAMS

	Exam	Date	Session
	GEOMETRIA II	12/04/2023 - 14:00	SESSIONE ORDINARIA

	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.

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. BILINEAR FORMS BILINEAR MAPS. SYMMETRIC AND SKEWSYMMETRIC BILINEAR FORMS. REPRESENTATIVE MATRIX. BILINEAR EXTENSION THEOREM. CHANGE OF FRAME. DEGENERATE BILINEAR FORMS. ANNIHILATOR SUBSPACES. QUADRATIC FORMS. ORTHOGONALITY. ORTHOGONAL BASES. EXISTENCE OF ORTHOGONAL BASES. NORMAL FORM OF A REAL SYMMETRIC BILINEAR FORM: SYLVESTER THEOREM. 2. 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. THE ADJOINT ENDOMORPHISM. ORTHOGONAL MAPS. THE ORTHOGONAL GROUP. THE SPECIAL ORTHOGONAL GROUP. CLASSIFICATION OF ORTHOGONAL TRANSFORMATIONS IN DIMENSION 2 AND 3. 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, ORIENTATIONS, 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.

Contents

1. BILINEAR FORMS

BILINEAR MAPS. SYMMETRIC AND SKEWSYMMETRIC BILINEAR FORMS. REPRESENTATIVE MATRIX. BILINEAR EXTENSION THEOREM. CHANGE OF FRAME. DEGENERATE BILINEAR FORMS. ANNIHILATOR SUBSPACES. QUADRATIC FORMS. ORTHOGONALITY. ORTHOGONAL BASES. EXISTENCE OF ORTHOGONAL BASES. NORMAL FORM OF A REAL SYMMETRIC BILINEAR FORM: SYLVESTER THEOREM.

2. 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. THE ADJOINT ENDOMORPHISM. ORTHOGONAL MAPS. THE ORTHOGONAL GROUP. THE SPECIAL ORTHOGONAL GROUP. CLASSIFICATION OF ORTHOGONAL TRANSFORMATIONS IN DIMENSION 2 AND 3.

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, ORIENTATIONS, 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
	64 HOURS OF LECTURES DIVIDED BETWEEN THEORETICAL LESSONS AND EXERCISES.

	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. THE MINIMUM MARK FOR THE ORAL EXAM REQUIRES KNOWLEDGE OF THE FUNDAMENTAL CONCEPTS OF THE TOPICS MENTIONED ABOVE. THE MAXIMUM MARK IS OBTAINED WHEN THE STUDENT SHOWS MASTERY IN THE DISCUSSION OF SUCH TOPICS.

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. THE MINIMUM MARK FOR THE ORAL EXAM REQUIRES KNOWLEDGE OF THE FUNDAMENTAL CONCEPTS OF THE TOPICS MENTIONED ABOVE. THE MAXIMUM MARK IS OBTAINED WHEN THE STUDENT SHOWS MASTERY IN THE DISCUSSION OF SUCH TOPICS.

	Texts
	LECTURE NOTES BY THE INSTRUCTOR. R. ESPOSITO, A. RUSSO, LEZIONI DI GEOMETRIA, PARTE PRIMA, LIGUORI. E. SERNESI, GEOMETRIA 1, BOLLATI BORINGHIERI. S. LIPSCHUTZ, ALGEBRA LINEARE MCGRAW-HILL.