Antonio DE NICOLA | GEOMETRY

cod. 0512600008

GEOMETRY

	0512600008
	DEPARTMENT OF PHYSICS "E. R. CAIANIELLO"
	EQF6
	PHYSICS
	2024/2025

PERCORSO SHARED STUDY PATHWAY

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2017
	SPRING SEMESTER

TEACHING UNITS

SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
MAT/03	8	64	LESSONS	SUPPLEMENTARY COMPULSORY SUBJECTS
MAT/03	1	12	EXERCISES	SUPPLEMENTARY COMPULSORY SUBJECTS

EXAMS

	Exam	Date	Session
	APPELLO DI APRILE 2025	14/04/2025 - 14:30	SESSIONE ORDINARIA

	Objectives
	THE AIM OF THE COURSE IS TO INTRODUCE STUDENTS TO THE THEORY OF VECTOR SPACES AND TO THE THEORY OF AFFINE AND EUCLIDEAN GEOMETRY. KNOWLEDGE AND UNDERSTANDING THE COURSE AIMS TO PROVIDE THE FUNDAMENTAL TOOLS OF LINEAR ALGEBRA WHICH, BESIDES THEIR GENERAL USEFULNESS IN THE STUDY OF PHYSICS, ARE ESSENTIAL FOR THE STUDY OF AFFINE GEOMETRY. WITH THE USE OF THESE TOOLS THE STUDENTS WILL BE INTRODUCED TO THE STUDY OF AFFINE AND EUCLIDEAN SPACES, AFFINE AND ISOMETRIC MAPS, AND CONIC SECTIONS. APPLYING KNOWLEDGE AND UNDERSTANDING THE COURSE AIMS TO ENABLE STUDENTS TO USE THE CALCULATION TOOLS RELATED TO THE ABOVE MENTIONED TOPICS. IN PARTICULAR, THE STUDENT WILL KNOW HOW TO OPERATE WITH MATRICES, SOLVING SYSTEMS OF LINEAR EQUATIONS AND DEALING WITH ISSUES RELATED TO VECTOR SPACES, LINEAR APPLICATIONS, AND AFFINE AND EUCLIDEAN SPACES WITH SPECIAL EMPHASIS ON SPACES OF DIMENSIONS TWO AND THREE.

THE AIM OF THE COURSE IS TO INTRODUCE STUDENTS TO THE THEORY OF VECTOR SPACES AND TO THE THEORY OF AFFINE AND EUCLIDEAN GEOMETRY.

KNOWLEDGE AND UNDERSTANDING
THE COURSE AIMS TO PROVIDE THE FUNDAMENTAL TOOLS OF LINEAR ALGEBRA WHICH, BESIDES THEIR GENERAL USEFULNESS IN THE STUDY OF PHYSICS, ARE ESSENTIAL FOR THE STUDY OF AFFINE GEOMETRY. WITH THE USE OF THESE TOOLS THE STUDENTS WILL BE INTRODUCED TO THE STUDY OF AFFINE AND EUCLIDEAN SPACES, AFFINE AND ISOMETRIC MAPS, AND CONIC SECTIONS.

APPLYING KNOWLEDGE AND UNDERSTANDING
THE COURSE AIMS TO ENABLE STUDENTS TO USE THE CALCULATION TOOLS RELATED TO THE ABOVE MENTIONED TOPICS. IN PARTICULAR, THE STUDENT WILL KNOW HOW TO OPERATE WITH MATRICES, SOLVING SYSTEMS OF LINEAR EQUATIONS AND DEALING WITH ISSUES RELATED TO VECTOR SPACES, LINEAR APPLICATIONS, AND AFFINE AND EUCLIDEAN SPACES WITH SPECIAL EMPHASIS ON SPACES OF DIMENSIONS TWO AND THREE.

	Prerequisites
	IT IS REQUIRED THAT STUDENTS HAVE A GOOD KNOWLEDGE OF THE BASIC TOPICS IN MATHEMATICS COVERED IN HIGH SCHOOL.

	Contents
	1. VECTOR SPACES, 10 HOURS LECTURES + 2 HOURS EXERCISES. LINEAR DEPENDENCE AND INDEPENDENCE. BASES, STEINITZ LEMMA, DIMENSION. SUBSPACES, SUMS AND DIRECT SUMS. GRASSMANN FORMULA. COORDINATE SYSTEMS. 2. MATRICES, DETERMINANTS AND SYSTEMS OF LINEAR EQUATIONS, 12 HOURS LECTURES + 4 HOURS EXERCISES. MATRICES, OPERATIONS ON MATRICES. ELEMENTARY OPERATIONS. MATRICES IN ROW ECHELON FORM AND GAUSS-JORDAN ALGORITHM, RANK. PERMUTATIONS. DETERMINANTS. LAPLACE THEOREM. KRONECKER THEOREM. BINET THEOREM (WITHOUT PROOF). INVERTIBLE MATRICES, COMPUTATION OF THE INVERSE MATRIX. SYSTEMS OF LINEAR EQUATIONS. RESOLUTION OF SYSTEMS OF LINEAR EQUATIONS IN ROW ECHELON FORM. REDUCTION OF A SOLVABLE SYSTEM OF LINEAR EQUATION TO ROW ECHELON FORM. ROUCHÉ-CAPELLI THEOREM. CRAMER'S THEOREM. 3. LINEAR MAPS, 9 HOURS LECTURES + 2 HOURS EXERCISES. DEFINITION, KERNEL AND IMAGE. LINEAR EXTENSION THEOREM. RANK–NULLITY THEOREM. MATRIX REPRESENTATION OF A LINEAR MAP. RANK OF A LINEAR MAP. PARAMETRIC AND CARTESIAN SUBSPACES REPRESENTATIONS. CHANGE OF FRAME. GENERAL LINEAR GROUP. 4. LINEAR AND BILINEAR FORMS, 5 HOURS LECTURES DUAL SPACE OF A VECTOR SPACE, DUAL BASES, ANNIHILATOR SUBSPACE OF A LINEAR SUBSPACE. BILINEAR MAPS, SYMMETRIC AND SKEW-SYMMETRIC BILINEAR FORMS. MATRIX REPRESENTATION OF BILINEAR FORMS. LINEAR EXTENSION THEOREM. CHANGE OF FRAME. DEGENERATE BILINEAR FORMS. ANNIHILATOR SUBSPACES. QUADRATIC FORMS. ORTHOGONALITY BETWEEN A VECTOR AND A LINEAR SUBSPACE. ORTHOGONAL BASES, EXISTENCE OF ORTHOGONAL BASES. CANONICAL FORM OF A BILINEAR FORM: SYLVESTER'S THEOREM. POSITIVE DEFINITE AND SEMIDEFINITE REAL SYMMETRIC BILINEAR FORMS. 5. EUCLIDEAN VECTOR SPACES, 4 HOURS LECTURES + 1 HOUR EXERCISES. SCALAR PRODUCTS. NORM AND ITS PROPERTIES. ANGLE BETWEEN TWO VECTORS. CAUCHY–SCHWARZ INEQUALITY. ORTHOGONALITY. GRAM-SCHMIDT ORTHOGONALIZATION PROCESS. COMPONENTS OF A VECTOR IN AN ORTHONORMAL BASIS. ORTHOGONAL MATRICES AND ORTHONORMAL NUMERICAL VECTORS. CHANGE OF ORTHONORMAL BASES. ORTHOGONAL SUBSPACES. ORTHOGONAL COMPLEMENT. ORTHOGONAL DECOMPOSITION. 6. THE PROBLEM OF DIAGONALIZATION, 7 HOURS LECTURES + 4 HOURS EXERCISES. DIAGONALIZATION OF AN ENDOMORPHISM. EIGENVALUES, EIGENVECTORS, EIGENSPACES. HOW TO FIND EIGENVALUES, CHARACTERISTIC POLYNOMIAL, ALGEBRAIC AND GEOMETRIC MULTIPLICITY. DIAGONALIZABILITY THEOREMS. DIAGONALIZATION OF ENDOMORPHISMS OF EUCLIDEAN VECTOR SPACES. ORTHOGONAL DIAGONALIZABILITY. SYMMETRIC ENDOMORPHISMS, SYMMETRIC MATRICES. EIGENVALUES OF A SYMMETRIC ENDOMORPHISM. ORTHOGONAL ENDOMORPHISMS AND THEIR REPRESENTATIONS. ORTHOGONAL MATRICES. EIGENVALUES OF AN ORTHOGONAL ENDOMORPHISM. ORTHOGONALLY DIAGONALIZABLE ENDOMORPHISMS. THE SPECTRAL THEOREM. 7. HERMITIAN FORMS, 4 HOURS LECTURES + 1 HOUR EXERCISES. HERMITIAN FORMS AND THEIR REPRESENTATIONS. HERMITIAN MATRICES. DIAGONALIZATION OF A HERMITIAN FORM. HERMITIAN FORMS CLASSIFICATION. HERMITIAN PRODUCTS. THE STANDARD HERMITIAN PRODUCT. HERMITIAN VECTOR SPACES. HERMITIAN OPERATORS. EIGENVALUES OF AN HERMITIAN OPERATOR. DIAGONALIZATION OF HERMITIAN OPERATORS. UNITARY OPERATORS. EIGENVALUES OF AN UNITARY OPERATOR. DIAGONALIZATION OF UNITARY OPERATORS. 8. AFFINE AND EUCLIDEAN AFFINE SPACES, 7 HOURS LECTURES + 4 HOURS EXERCISES. AFFINE SPACES. AFFINE SUBSPACES. AFFINE FRAMES. AFFINE SUBSPACES REPRESENTATIONS. PARALLELISM AND INTERSECTION OF SUBSPACES. GEOMETRY IN AN AFFINE SPACE OF DIMENSION 2 AND 3. EUCLIDEAN AFFINE SPACES, CARTESIAN FRAMES, DISTANCE BETWEEN TWO POINTS, ANGLE BETWEEN TWO LINES. GEOMETRY IN AN EUCLIDEAN AFFINE SPACE OF DIMENSION 2 E 3. AFFINITIES. ISOMETRIES. INTRODUCTION TO EUCLIDEAN CONIC SECTIONS.

Contents

1. VECTOR SPACES, 10 HOURS LECTURES + 2 HOURS EXERCISES.
LINEAR DEPENDENCE AND INDEPENDENCE. BASES, STEINITZ LEMMA, DIMENSION. SUBSPACES, SUMS AND DIRECT SUMS. GRASSMANN FORMULA. COORDINATE SYSTEMS.

2. MATRICES, DETERMINANTS AND SYSTEMS OF LINEAR EQUATIONS, 12 HOURS LECTURES + 4 HOURS EXERCISES.
MATRICES, OPERATIONS ON MATRICES. ELEMENTARY OPERATIONS. MATRICES IN ROW ECHELON FORM AND GAUSS-JORDAN ALGORITHM, RANK. PERMUTATIONS. DETERMINANTS. LAPLACE THEOREM. KRONECKER THEOREM. BINET THEOREM (WITHOUT PROOF). INVERTIBLE MATRICES, COMPUTATION OF THE INVERSE MATRIX. SYSTEMS OF LINEAR EQUATIONS. RESOLUTION OF SYSTEMS OF LINEAR EQUATIONS IN ROW ECHELON FORM. REDUCTION OF A SOLVABLE SYSTEM OF LINEAR EQUATION TO ROW ECHELON FORM. ROUCHÉ-CAPELLI THEOREM. CRAMER'S THEOREM.

3. LINEAR MAPS, 9 HOURS LECTURES + 2 HOURS EXERCISES.
DEFINITION, KERNEL AND IMAGE. LINEAR EXTENSION THEOREM. RANK–NULLITY THEOREM. MATRIX REPRESENTATION OF A LINEAR MAP. RANK OF A LINEAR MAP. PARAMETRIC AND CARTESIAN SUBSPACES REPRESENTATIONS. CHANGE OF FRAME. GENERAL LINEAR GROUP.

4. LINEAR AND BILINEAR FORMS, 5 HOURS LECTURES
DUAL SPACE OF A VECTOR SPACE, DUAL BASES, ANNIHILATOR SUBSPACE OF A LINEAR SUBSPACE. BILINEAR MAPS, SYMMETRIC AND SKEW-SYMMETRIC BILINEAR FORMS. MATRIX REPRESENTATION OF BILINEAR FORMS. LINEAR EXTENSION THEOREM. CHANGE OF FRAME. DEGENERATE BILINEAR FORMS. ANNIHILATOR SUBSPACES. QUADRATIC FORMS. ORTHOGONALITY BETWEEN A VECTOR AND A LINEAR SUBSPACE. ORTHOGONAL BASES, EXISTENCE OF ORTHOGONAL BASES. CANONICAL FORM OF A BILINEAR FORM: SYLVESTER'S THEOREM. POSITIVE DEFINITE AND SEMIDEFINITE REAL SYMMETRIC BILINEAR FORMS.

5. EUCLIDEAN VECTOR SPACES, 4 HOURS LECTURES + 1 HOUR EXERCISES.
SCALAR PRODUCTS. NORM AND ITS PROPERTIES. ANGLE BETWEEN TWO VECTORS. CAUCHY–SCHWARZ INEQUALITY. ORTHOGONALITY. GRAM-SCHMIDT ORTHOGONALIZATION PROCESS. COMPONENTS OF A VECTOR IN AN ORTHONORMAL BASIS. ORTHOGONAL MATRICES AND ORTHONORMAL NUMERICAL VECTORS. CHANGE OF ORTHONORMAL BASES. ORTHOGONAL SUBSPACES. ORTHOGONAL COMPLEMENT. ORTHOGONAL DECOMPOSITION.

6. THE PROBLEM OF DIAGONALIZATION, 7 HOURS LECTURES + 4 HOURS EXERCISES.
DIAGONALIZATION OF AN ENDOMORPHISM. EIGENVALUES, EIGENVECTORS, EIGENSPACES. HOW TO FIND EIGENVALUES, CHARACTERISTIC POLYNOMIAL, ALGEBRAIC AND GEOMETRIC MULTIPLICITY. DIAGONALIZABILITY THEOREMS. DIAGONALIZATION OF ENDOMORPHISMS OF EUCLIDEAN VECTOR SPACES. ORTHOGONAL DIAGONALIZABILITY. SYMMETRIC ENDOMORPHISMS, SYMMETRIC MATRICES. EIGENVALUES OF A SYMMETRIC ENDOMORPHISM. ORTHOGONAL ENDOMORPHISMS AND THEIR REPRESENTATIONS. ORTHOGONAL MATRICES. EIGENVALUES OF AN ORTHOGONAL ENDOMORPHISM. ORTHOGONALLY DIAGONALIZABLE ENDOMORPHISMS. THE SPECTRAL THEOREM.

7. HERMITIAN FORMS, 4 HOURS LECTURES + 1 HOUR EXERCISES.
HERMITIAN FORMS AND THEIR REPRESENTATIONS. HERMITIAN MATRICES. DIAGONALIZATION OF A HERMITIAN FORM. HERMITIAN FORMS CLASSIFICATION. HERMITIAN PRODUCTS. THE STANDARD HERMITIAN PRODUCT. HERMITIAN VECTOR SPACES. HERMITIAN OPERATORS. EIGENVALUES OF AN HERMITIAN OPERATOR. DIAGONALIZATION OF HERMITIAN OPERATORS. UNITARY OPERATORS. EIGENVALUES OF AN UNITARY OPERATOR. DIAGONALIZATION OF UNITARY OPERATORS.

8. AFFINE AND EUCLIDEAN AFFINE SPACES, 7 HOURS LECTURES + 4 HOURS EXERCISES.
AFFINE SPACES. AFFINE SUBSPACES. AFFINE FRAMES. AFFINE SUBSPACES REPRESENTATIONS. PARALLELISM AND INTERSECTION OF SUBSPACES. GEOMETRY IN AN AFFINE SPACE OF DIMENSION 2 AND 3. EUCLIDEAN AFFINE SPACES, CARTESIAN FRAMES, DISTANCE BETWEEN TWO POINTS, ANGLE BETWEEN TWO LINES. GEOMETRY IN AN EUCLIDEAN AFFINE SPACE OF DIMENSION 2 E 3. AFFINITIES. ISOMETRIES. INTRODUCTION TO EUCLIDEAN CONIC SECTIONS.

	Teaching Methods
	76 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 LASTS THREE HOURS. IT CONSISTS OF SOME EXERCISES. THE ORAL EXAM EVALUATES THE ACQUIRED KNOWLEDGE OF LINEAR ALGEBRA, THE THEORY OF AFFINE AND EUCLIDEAN SPACES. THE FINAL GRADE IS EXPRESSED IN THIRTIETHS. PASSING THE WRITTEN TEST GIVES ACCESS TO THE ORAL TEST, WHICH DETERMINES THE FINAL GRADE IN FULL. TO PASS THE EXAM WITH THE MINIMUM GRADE, OF 18/30, ONE MUST SHOW MASTERY OF THE FUNDAMENTAL CONCEPTS. TO PASS THE TEST WITH 30/30 ONE MUST SHOW MASTERY AND CONFIDENCE ON BOTH THE FUNDAMENTAL CONCEPTS AND THE DETAILS OF THE DEMONSTRATIONS. TO OBTAIN HONORS, ONE MUST SHOW A PARTICULARLY LUCID VIEW OF BOTH THE DETAIL AND THE WHOLE. PASSING ONLY THE WRITTEN TEST IN ONE EXAM SESSION EXEMPTS ONE FROM HAVING TO RETAKE IT TO GAIN ACCESS TO THE ORAL WITHIN THE NEXT TWO EXAM 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 LASTS THREE HOURS. IT CONSISTS OF SOME EXERCISES. THE ORAL EXAM EVALUATES THE ACQUIRED KNOWLEDGE OF LINEAR ALGEBRA, THE THEORY OF AFFINE AND EUCLIDEAN SPACES.
THE FINAL GRADE IS EXPRESSED IN THIRTIETHS. PASSING THE WRITTEN TEST GIVES ACCESS TO THE ORAL TEST, WHICH DETERMINES THE FINAL GRADE IN FULL. TO PASS THE EXAM WITH THE MINIMUM GRADE, OF 18/30, ONE MUST SHOW MASTERY OF THE FUNDAMENTAL CONCEPTS. TO PASS THE TEST WITH 30/30 ONE MUST SHOW MASTERY AND CONFIDENCE ON BOTH THE FUNDAMENTAL CONCEPTS AND THE DETAILS OF THE DEMONSTRATIONS. TO OBTAIN HONORS, ONE MUST SHOW A PARTICULARLY LUCID VIEW OF BOTH THE DETAIL AND THE WHOLE.
PASSING ONLY THE WRITTEN TEST IN ONE EXAM SESSION EXEMPTS ONE FROM HAVING TO RETAKE IT TO GAIN ACCESS TO THE ORAL WITHIN THE NEXT TWO EXAM SESSIONS.

	Texts
	MAIN COURSE BOOK E. SERNESI, "GEOMETRIA 1", BOLLATI BORINGHIERI. REFERENCE BOOKS B. MARTELLI, GEOMETRIA E ALGEBRA LINEARE, AUTOPUBBLICATO, GRATUITO ONLINE. PH. ELLIA, "APPUNTI DI GEOMETRIA I". PITAGORA ED. EXERCISE BOOK S. LIPSCHUTZ, "ALGEBRA LINEARE", MCGRAW-HILL.

	More Information
	TEACHER'S E-MAIL ADDRESS: ANDENICOLA@UNISA.IT

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Antonio DE NICOLA | GEOMETRY

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