# Fisica | GEOMETRY

## Fisica GEOMETRY

 0512600008 DEPARTMENT OF PHYSICS "E. R. CAIANIELLO" EQF6 PHYSICS 2022/2023

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2017 SPRING SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/03 9 72 LESSONS SUPPLEMENTARY COMPULSORY SUBJECTS
 ANTONIO DE NICOLA T
ExamDate
APPELLO DI FEBBRAIO 202313/02/2023 - 10:00
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 THEIR PHYSICAL APPLICATIONS.
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, 5 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 AN 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. ORTOGONALMENTE 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
72 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 LINEAR ALGEBRA, THE THEORY OF AFFINE SPACES AND EUCLIDEAN CONICS.
THE FINAL EVALUATION 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. PASSING THE WRITTEN EXAM IN ONE OF THE SESSIONS WILL EXONERATE FROM THE WRITTEN EXAM UNTIL SEPTEMBER.
Texts
E. SERNESI, "GEOMETRIA 1", BOLLATI BORINGHIERI.
R. ESPOSITO, A. RUSSO, "LEZIONI DI GEOMETRIA, PARTE PRIMA", LIGUORI.
S. LIPSCHUTZ, "ALGEBRA LINEARE", MCGRAW-HILL.