Luca VITAGLIANO | GEOMETRY II

cod. MT12300040

GEOMETRY II

	MT12300040
	DEPARTMENT OF MATHEMATICS
	EQF6
	MATHEMATICS
	2025/2026

PERCORSO STANDARD EDUCATIONAL PATH

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2025
	SPRING SEMESTER

TEACHING UNITS

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

	Objectives
	THIS COURSE AIMS AT INTRODUCING THE STUDENTS TO THE THEORY OF EUCLIDEAN VECTOR SPACES AND TO AFFINE AND EUCLIDEAN GEOMETRY. KNOWLEDGE AND UNDERSTANDING: THE COURSE MEANS TO PROVIDE THE STUDENTS WITH A SOLID KNOWLEDGE OF EUCLIDEAN VECTOR SPACES, AFFINE SPACES AND AFFINE AND ISOMETRIC MAPS. AT THE END OF THE COURSE THE STUDENT WILL BE ABLE TO UNDERSTAND THE FUNDAMENTALS OF EUCLIDEAN GEOMETRY FROM THE POINT OF VIEW OF MODERN MATHEMATICS. APPLYING KNOWLEDGE AND UNDERSTANDING: THE COURSE HAS THE FURTHER AIM TO ENABLE THE STUDENT TO SOLVE PROBLEMS ON EUCLIDEAN VECTOR SPACES AND AFFINE SPACES, WITH A SPECIAL EMPHASIS ON 2-DIMENSIONAL AND 3-DIMENSIONAL SPACES. AUTONOMY OF JUDGEMENT AT THE END OF THE COURSE THE STUDENT WILL BE ABLE TO DISTINGUISH THE GEOMETRIC NATURE OF THE SIMPLEST LINEAR ALGEBRA PROBLEMS HE MIGHT ENCOUNTER, INCLUDING THOSE IN INTERDISCIPLINARY FIELDS, WITH A SPECIAL EMPHASIS ON SYSTEMS OF LINEAR EQUATIONS. THEY WILL ALSO BE ABLE TO APPLY THE GEOMETRIC METHOD TO THOSE PROBLEMS AND, IN PARTICULAR, TO ASSESS WHICH ARE THE ASPECTS OF THE PROBLEM THAT DO NOT DEPEND ON THE CHOICE OF COORDINATES. COMMUNICATION SKILLS DURING THE EXERCISE SESSIONS AND PREPARING THE WRITTEN TEST, THE STUDENT WILL LEARN HOW TO COMMUNICATE BOTH ORALLY AND IN WRITING, IN A CLEAR AND CONCISE WAY, THE SOLUTIONS OF THE EXERCISES THAT THEY HAVE INDEPENDENTLY SOLVED. MOREOVER, PREPARING THE ORAL TEST, THE STUDENT WILL LEARN HOW TO EXPRESS IN ORGANIC AND COMPLETE WAY, AND USING THE APPROPRIATE “LITERATE REGISTER”, NOT ONLY THE SINGLE DEFINITIONS AND PROPOSITIONS DISCUSSED DURING THE LECTURES, BUT ALSO THE LOGICAL CONNECTIONS BETWEEN DIFFERENT PARTS OF THE PROGRAM AND, MORE GENERALLY, THE STRUCTURE OF THE THEORY. LEARNING SKILLS AT THE END OF THE COURSE THE STUDENT WILL BE ABLE TO CARRY OUT BIBLIOGRAPHIC SEARCHES AND TO STUDY ON THEIR OWN ADVANCED TOPICS IN EUCLIDEAN AND AFFINE GEOMETRY, INCLUDING THOSE OF AN INTERDISCIPLINARY NATURE WHICH HAVE NOT BEEN DISCUSSED DURING THE COURSE.

THIS COURSE AIMS AT INTRODUCING THE STUDENTS TO THE THEORY OF EUCLIDEAN VECTOR SPACES AND TO AFFINE AND EUCLIDEAN GEOMETRY.

KNOWLEDGE AND UNDERSTANDING:

THE COURSE MEANS TO PROVIDE THE STUDENTS WITH A SOLID KNOWLEDGE OF EUCLIDEAN VECTOR SPACES, AFFINE SPACES AND AFFINE AND ISOMETRIC MAPS. AT THE END OF THE COURSE THE STUDENT WILL BE ABLE TO UNDERSTAND THE FUNDAMENTALS OF EUCLIDEAN GEOMETRY FROM THE POINT OF VIEW OF MODERN MATHEMATICS.

APPLYING KNOWLEDGE AND UNDERSTANDING:

THE COURSE HAS THE FURTHER AIM TO ENABLE THE STUDENT TO SOLVE PROBLEMS ON EUCLIDEAN VECTOR SPACES AND AFFINE SPACES, WITH A SPECIAL EMPHASIS ON 2-DIMENSIONAL AND 3-DIMENSIONAL SPACES.

AUTONOMY OF JUDGEMENT

AT THE END OF THE COURSE THE STUDENT WILL BE ABLE TO DISTINGUISH THE GEOMETRIC NATURE OF THE SIMPLEST LINEAR ALGEBRA PROBLEMS HE MIGHT ENCOUNTER, INCLUDING THOSE IN INTERDISCIPLINARY FIELDS, WITH A SPECIAL EMPHASIS ON SYSTEMS OF LINEAR EQUATIONS. THEY WILL ALSO BE ABLE TO APPLY THE GEOMETRIC METHOD TO THOSE PROBLEMS AND, IN PARTICULAR, TO ASSESS WHICH ARE THE ASPECTS OF THE PROBLEM THAT DO NOT DEPEND ON THE CHOICE OF COORDINATES.

COMMUNICATION SKILLS

DURING THE EXERCISE SESSIONS AND PREPARING THE WRITTEN TEST, THE STUDENT WILL LEARN HOW TO COMMUNICATE BOTH ORALLY AND IN WRITING, IN A CLEAR AND CONCISE WAY, THE SOLUTIONS OF THE EXERCISES THAT THEY HAVE INDEPENDENTLY SOLVED. MOREOVER, PREPARING THE ORAL TEST, THE STUDENT WILL LEARN HOW TO EXPRESS IN ORGANIC AND COMPLETE WAY, AND USING THE APPROPRIATE “LITERATE REGISTER”, NOT ONLY THE SINGLE DEFINITIONS AND PROPOSITIONS DISCUSSED DURING THE LECTURES, BUT ALSO THE LOGICAL CONNECTIONS BETWEEN DIFFERENT PARTS OF THE PROGRAM AND, MORE GENERALLY, THE STRUCTURE OF THE THEORY.

LEARNING SKILLS

AT THE END OF THE COURSE THE STUDENT WILL BE ABLE TO CARRY OUT BIBLIOGRAPHIC SEARCHES AND TO STUDY ON THEIR OWN ADVANCED TOPICS IN EUCLIDEAN AND AFFINE GEOMETRY, INCLUDING THOSE OF AN INTERDISCIPLINARY NATURE WHICH HAVE NOT BEEN DISCUSSED DURING THE COURSE.

	Prerequisites
	THE STUDENT MUST HAVE SUCCESSFULLY PASSED THE GEOMETRY I EXAM.

	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. THEORETICAL LECTURES 9 HRS, EXERCISES 2 HRS. 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. THEORETICAL LECTURES 13 HRS, EXERCISES 3 HRS 3. HERMITIAN FORMS HERMITEAN FORMS AND REPRESENTATIONS. HERMITEAN MATRICES. HERMITEAN PRODUCTS. THE STANDARD HERMITEAN PRODUCT. HERMITEAN SPACES. UNITARY AND HERMITEAN ENDOMORPHISMS. THEORETICAL LECTURES 9 HRS, EXERCISES 2 HRS 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. THEORETICAL LECTURES 17 HRS, EXERCISES 3 HRS 5. QUADRICS AND CONICS QUADRICS. NORMAL FORM OF REAL QUADRICS. CONICS. THEORETICAL LECTURES 4 HRS, EXERCISES 2 HRS.

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.

THEORETICAL LECTURES 9 HRS, EXERCISES 2 HRS.

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.

THEORETICAL LECTURES 13 HRS, EXERCISES 3 HRS

3. HERMITIAN FORMS

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

THEORETICAL LECTURES 9 HRS, EXERCISES 2 HRS

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.

THEORETICAL LECTURES 17 HRS, EXERCISES 3 HRS

5. QUADRICS AND CONICS

QUADRICS. NORMAL FORM OF REAL QUADRICS. CONICS.

THEORETICAL LECTURES 4 HRS, EXERCISES 2 HRS.

	Teaching Methods
	64 HOURS OF LECTURES DIVIDED BETWEEN THEORETICAL LESSONS AND EXERCISES.

	Verification of learning
	THE EXAM CONSISTS IN A WRITTEN AND AN ORAL TEST. IN THE WRITTEN TEST (DURATION AROUND 2H30M) THE STUDENT WILL HAVE TO SOLVE 2-3 EXERCISES AIMING AT CHECKING THE ABILITY OF THE STUDENT TO APPLY THE METHODS OF EUCLIDEAN VECTOR SPACES AND AFFINE SPACES. THE ORAL TEST (DURATION AROUND 30M) AIMS AT ASSESSING THE THEORETICAL KNOWLEDGE OF THE STUDENT AND THE ABILITY TO INDEPENDENTLY DEVELOP THE LINES OF REASONING PRESENTED DURING THE COURSE. THE WRITTEN TEST REQUIRES THE APPLICATION OF THE METHODS OF EUCLIDEAN VECTOR SPACES AND AFFINE SPACES, IN PARTICULAR THE DIAGONALIZATION OF A SYMMETRIC ENDOMORPHISM OROF A SYMMETRIC BILINEAR FORM, AND THE SOLUTION TO A PROBLEM IN 2D OR 3D AFFINE GEOMETRY. A POSSIBLE THIRD EXERCISE MIGHT REQUIRE PROVING AN ELEMENTARY ASSERTION WHICH HAVE NOT BEEN PREVIOUSLY DISCUSSED DURING THE LECTURES, TO CHECK THE AUTONOMY OF JUDGEMENT AND THE ABILITY TO APPLY THE THEORETICAL TOOLS PROVIDED IN THE LECTURES. THE ORAL TEST IS ACCESSIBLE TO THOSE STUDENTS WHO SUCCESSFULLY PASS THE WRITTEN TEST. THE MARKS AT THE WRITTEN TEST DO NOT DETERMINE THE FINAL MARKS. THE MARKS AT THE ORAL TEST DEPEND ON THE DEPTH OF KNOWLEDGE AND THE STUDENT'S ABILITY TO PRESENT THE COURSE CONTENTS IN AN EFFICIENT WAY, AND TO CRITICALLY DISCUSS THE TOPICS ILLUSTRATED DURING THE COURSE. THE MINIMAL MARKS IN THE ORAL TEST REQUIRE THE KNOWLEDGE OF THE FOLLOWING TOPICS: BILINEAR FORMS, EUCLIDEAN AND HERMITEAN VECTOR SPACES, AFFINE SPACES AND THE CLASSIFICATION OF QUADRICS. THE MAXIMUM MARKS ARE GRANTED WHEN THE STUDENT SHOWS THEY KNOW HOW TO ILLUSTRATE THE ABOVE TOPICS WITH A HIGH DEGREE OF COMPETENCE AND CONFIDENCE.

Verification of learning

THE EXAM CONSISTS IN A WRITTEN AND AN ORAL TEST. IN THE WRITTEN TEST (DURATION AROUND 2H30M) THE STUDENT WILL HAVE TO SOLVE 2-3 EXERCISES AIMING AT CHECKING THE ABILITY OF THE STUDENT TO APPLY THE METHODS OF EUCLIDEAN VECTOR SPACES AND AFFINE SPACES. THE ORAL TEST (DURATION AROUND 30M) AIMS AT ASSESSING THE THEORETICAL KNOWLEDGE OF THE STUDENT AND THE ABILITY TO INDEPENDENTLY DEVELOP THE LINES OF REASONING PRESENTED DURING THE COURSE.

THE WRITTEN TEST REQUIRES THE APPLICATION OF THE METHODS OF EUCLIDEAN VECTOR SPACES AND AFFINE SPACES, IN PARTICULAR THE DIAGONALIZATION OF A SYMMETRIC ENDOMORPHISM OROF A SYMMETRIC BILINEAR FORM, AND THE SOLUTION TO A PROBLEM IN 2D OR 3D AFFINE GEOMETRY. A POSSIBLE THIRD EXERCISE MIGHT REQUIRE PROVING AN ELEMENTARY ASSERTION WHICH HAVE NOT BEEN PREVIOUSLY DISCUSSED DURING THE LECTURES, TO CHECK THE AUTONOMY OF JUDGEMENT AND THE ABILITY TO APPLY THE THEORETICAL TOOLS PROVIDED IN THE LECTURES.

THE ORAL TEST IS ACCESSIBLE TO THOSE STUDENTS WHO SUCCESSFULLY PASS THE WRITTEN TEST. THE MARKS AT THE WRITTEN TEST DO NOT DETERMINE THE FINAL MARKS. THE MARKS AT THE ORAL TEST DEPEND ON THE DEPTH OF KNOWLEDGE AND THE STUDENT'S ABILITY TO PRESENT THE COURSE CONTENTS IN AN EFFICIENT WAY, AND TO CRITICALLY DISCUSS THE TOPICS ILLUSTRATED DURING THE COURSE. THE MINIMAL MARKS IN THE ORAL TEST REQUIRE THE KNOWLEDGE OF THE FOLLOWING TOPICS: BILINEAR FORMS, EUCLIDEAN AND HERMITEAN VECTOR SPACES, AFFINE SPACES AND THE CLASSIFICATION OF QUADRICS. THE MAXIMUM MARKS ARE GRANTED WHEN THE STUDENT SHOWS THEY KNOW HOW TO ILLUSTRATE THE ABOVE TOPICS WITH A HIGH DEGREE OF COMPETENCE AND CONFIDENCE.

	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.

	More Information
	EMAIL: lvitagliano@unisa.it

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Luca VITAGLIANO | GEOMETRY II

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Luca VITAGLIANO | GEOMETRY II