# Ingegneria Edile-Architettura | GEOMETRY

## Ingegneria Edile-Architettura GEOMETRY

 0660100002 DEPARTMENT OF CIVIL ENGINEERING EQF7 BUILDING ENGINEERING - ARCHITECTURE 2022/2023

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2017 AUTUMN SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/03 6 60 LESSONS BASIC COMPULSORY SUBJECTS
 GIOVANNINA ALBANO T ANNA PIERRI
ExamDate
GEOMETRIA09/05/2023 - 14:00
Objectives
EXPECTED LEARNING OUTCOMES AND COMPETENCE TO BE ACQUIRED:
TO ACQUIRE THE BASIC ELEMENTS OF LINEAR ALGEBRA AND ANALYTICAL GEOMETRY: MATRICES, LINEAR SYSTEMS, VECTOR SPACES, EUCLIDEAN SPACES, LINEAR APPLICATIONS, DIAGONALIZATION, ANALYTICAL GEOMETRY IN 2D AND 3D. ACQUIRE THE RESULTS AND PROOF TECHNIQUES, AS WELL AS THE ABILITY TO USE THE RELEVANT COMPUTATIONAL TOOLS. TO CONSOLIDATE BASIC MATHEMATICAL KNOWLEDGE; TO PROVIDE AND TO DEVELOP TOOLS FOR A SCIENTIFIC APPROACH TO PROBLEMS AND PHENOMENA THAT THE STUDENT WILL FACE IN THE CONTINUATION OF HIS/HER STUDIES.

KNOWLEDGE AND UNDERSTANDING SKILLS:
TO KNOW AND UNDERSTAND THE BASIC CONCEPTS AND TERMINOLOGY USED IN LINEAR ALGEBRA AND ANALYTICAL GEOMETRY, RESULTS AND PROOF TECHNIQUES, AS WELL AS IN THE ABILITY TO USE THE RELEVANT COMPUTATIONAL TOOLS, WITH PARTICULAR REFERENCE TO THE FOLLOWING TOPICS: MATRICES, LINEAR SYSTEMS, VECTOR SPACES, EUCLIDEAN SPACES, LINEAR APPLICATIONS, DIAGONALIZATION, ANALYTICAL GEOMETRY IN PLANE AND SPACE.

ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
TO KNOW HOW TO APPLY THE STUDIED DEFINITIONS, THEOREMS AND RULES TO PROBLEM SOLVING; TO KNOW HOW TO DEVELOP IN A COHERENT WAY THE VARIOUS PROOFS; TO KNOW HOW TO USE LINEAR ALGEBRA STRUCTURES AND TOOLS FOR THE MANAGEMENT AND MATHEMATICAL PROBLEM SOLVING; TO WORK WITH 2- AND 3-DIMENSIONAL OBJECTS; TO KNOW HOW TO COORDINATE DIFFERENT SEMIOTIC REPRESENTATIONS OF THE SAME MATHEMATICAL OBJECT.

AUTONOMY OF JUDGMENT:
TO KNOW HOW TO IDENTIFY THE MOST APPROPRIATE METHODS TO EFFICIENTLY SOLVE A MATHEMATICAL PROBLEM AND TO BE ABLE TO FIND OPTIMIZATIONS TO THE PROCESS OF SOLVING A MATHEMATICAL PROBLEM.

COMMUNICATION SKILLS:
TO KNOW HOW TO EXPOSE ORALLY A TOPIC RELATED TO LINEAR ALGEBRA AND ANALYTICAL GEOMETRY IN 2D AND 3D, WITH LANGUAGE PROPERTIES AND ABILITY TO COORDINATE DIFFERENT SYSTEMS OF SEMIOTIC REPRESENTATION (VERBAL, SYMBOLIC, FIGURAL).

ABILITY TO LEARN:
TO KNOW HOW TO APPLY THE ACQUIRED KNOWLEDGE TO CONTEXTS DIFFERENT FROM THOSE PRESENTED DURING THE COURSE AND TO DEEPEN THE TOPICS TREATED USING MATERIALS DIFFERENT FROM THOSE PROPOSED.
Prerequisites
FOR A PROFITABLE ACHIEVEMENT OF THE EDUCATIONAL GOALS THE STUDENT IS REQUIRED TO MASTER KNOWLEDGE CONCERNING BASIC MATHEMATICS, WITH PARTICULAR REFERENCE TO THE FOLLOWING TOPICS: SETS AND OPERATIONS, REPRESENTATION OF REAL NUMBERS AND OPERATION, EQUATIONS AND DISEQUATION OF FIRST AND SECOND ORDER, FUNCTIONS AND PROPERTIES, TRIGONOMETRY.
PROPEDEUTICITY: NOTHING.
Contents
MATRICES. DEFINITIONS AND PROPERTIES. DETERMINANT AND ITS COMPUTATION: LAPLACE’S FORMULA. RANK OF A MATRIX. KRONECKER’S THEOREM. ROW ECHELON MATRIX. INVERSE OF A MATRIX. (LECTURES HRS 3, PRACTICE HRS 4).

LINEAR SYSTEMS OF EQUATIONS. DEFINITIONS, MATRIX EQUATION, CONSISTENCY AND INCONSISTENCY, NUMBER OF SOLUTIONS. ROUCHÉ-CAPELLI THEOREM. CRAMER THEOREM AND RULE. GAUSSIAN ELIMINATION ALGORITHM. BASIS OF THE SOLUTIONS SET OF A HOMOGENEOUS LINEAR SYSTEM. DISCUSSION OF LINEAR SYSTEMS WITH A PARAMETER. (LECTURES HRS 4, PRACTICE HRS 4).

VECTOR SPACES. THE VECTOR SPACE STRUCTURE. LINEAR DEPENDENCE AND INDEPENDENCE OF VECTORS. GENERATORS. BASES. STEINITIZ’ LEMMA. DIMENSION OF A VECTOR SPACE AND RELATED THEOREM. VECTOR SUBSPACES AND THEIR INTERSECTION, SUM AND DIRECT SUM. GRASSMAN’S RULE. (LECTURES HRS 5, PRACTICE HRS 4).

EUCLIDEAN SPACES. DEFINITION OF DOT PRODUCT. DEFINITION OF REAL EUCLIDEAN VECTOR SPACE. DEFINITION OF NORM. CAUCHY-SCHWARZ INEQUALITY. DEFINITION OF ANGLE. DEFINITION OF ORTHOGONAL VECTORS. ORTHONORMAL BASES. COMPONENT IN AN ORTHONORMAL BASIS. ORTHOGONAL PROJECTIONS. GRAM-SCHMIDT PROCEDURE. (LECTURES HRS 4, PRACTICE HRS 3).

LINEAR OPERATORS. DEFINITIONS OF LINEAR OPERATORS (HOMOMORPHISM), ENDO-, MONO-, EPI- MORPHISM. KERNEL AND IMAGE: PROPERTIES AND CHARACTERIZATIONS. RANK-NULLITY THEOREM. MATRIX REPRESENTATION. CHANGE OF BASIS. 2D LINEAR OPERATORS AND THEIR GEOMETRIC INTERPRETATION (ROTATIONS, REFLECTIONS, DILATATIONS AND CONTRACTIONS, DEFORMATIONS. (LECTURES HRS 3, PRACTICE HRS 3).

DIAGONALIZATION. EIGENVALUES AND EIGENVECTORS: DEFINITIONS, CHARACTERISTIC POLYNOMIAL AND EQUATION. ALGEBRAIC AND GEOMETRIC MULTEPLICITIES. DEFINITION OF DIAGONALIZATION AND ITS CHARACTERIZATIONS (FOR MATRICES AND ENDOMORPHISMS). MAIN THEOREM. SUFFICIENT CONDITION FOR THE DIAGONALIZATION. DEFINITION AND CHARACTERIZATION OF SYMMETRIC ENDOMORPHISMS. PROPERTIES OF THE EIGENVALUES OF SYMMETRIC MATRICES. SPECTRAL THEOREM. (LECTURES HRS 4, PRACTICE HRS 3).

2D ANALYTICAL GEOMETRY. PLANE CARTESIAN COORDINATE SYSTEM. EQUATION OF A LINE (CARTESIAN, PARAMETRIC, SEGMENTAL). PARALLELISM OF LINES. PROPER AND IMPROPER BUNDLE OF STRAIGHT LINES. LINE THROUGH A POINT. ANGLE BETWEEN TWO LINES. CONDITIONS FOR PERPENDICULAR LINES. DISTANCES AND SYMMETRIES IN THE PLANE. CONICS: CLASSIFICATION AND CANONICAL FORMS. ALGORITHMS FOR REDUCING CONICS TO CANONICAL FORM. CHARACTERISTIC GEOMETRIC PARAMETERS. (LECTURES HRS 3, PRACTICE HRS 4).

3D ANALITYC GEOMETRY. SPATIAL CARTESIAN COORDINATE SYSTEM. EQUATION OF THE PLANE (CARTESIAN AND PARAMETRIC). EQUATION OF A LINE (CARTESIAN, PARAMETRIC, SEGMENTAL). BUNDLE AND STARS OF PLANS. CONDITIONS FOR PARALLEL AND PERPENDICULAR LINES, LINE AND PLANE, PLANS. NON COPLANAR LINES. ANGLE BETWEEN TWO PLANS, TWO LINE, A LINE AND A PLANE. DISTANCES AND SYMMETRIES IN THE SPACE. PROBLEMS ON PARALLELISM AND PERPENDICULARITY IN THE SPACE. QUADRICS: CLASSIFICATION AND CANONICAL FORMS. ALGORITHMS FOR REDUCING QUADRICS TO CANONICAL FORM. CHARACTERISTIC GEOMETRIC PARAMETERS. (LECTURES HRS 4, PRACTICE HRS 5).
Teaching Methods
THE COURSE CONSISTS OF 30 HOURS OF THEORETIC FRONTAL LECTURES WITH EXAMPLES AND 30 HOURS OF EXCERCISE SESSIONS, IN TOTAL 60 HOURS (6 CREDITS).
THE ATTENDANCE IS MANDATORY. THE STUDENT IS REQUIRED TO ATTEND AT LEAST THE 70% OF THE COURSE, USING HIS PERSONAL BADGE.
Verification of learning
LEARNING ASSESSMENT WILL BE DONE THROUGH A WRITTEN AND ORAL EXAM AT THE END OF THE COURSE. MIDTERM WRITTEN PROOFS COULD BE DONE DURING THE COURSE OR A PRE-EXAM JUST AT THE END OF THE COURSE.
THE WRITTEN EXAM CONSISTS OF EXERCISES OR PROBLEMS FOR EVALUATING THE KNOWLEDGE APPLICATION CAPABILITY RELATED TO THE COMPUTATION OF THE TYPICAL ELEMENT OF THE VECTOR AND EUCLIDEAN SPACES, TO THE COMPUTATION OF EIGENVALUES AND EIGENSPACES AND RELATED DIAGONALIZATION OF MATRICES AND ENDOMORPHISM, TO THE CLASSIFICATION AND REPRESENTATION OF LINES AND CONICS IN 2D SPACE AND OF LINES AND PLANES IN 3D SPACE AND TO THE COMPUTATION OF THE RELATED GEOMETRIC ELEMENTS.
THE ORAL EXAM, WHICH MAY INCLUDE EXERCISES, CONSISTS OF QUESTIONS ON THE SAME SUBJECTS AND SERVES TO EVALUATE THE LEVEL OF STUDENT’S THEORETICAL KNOWLEDGE, MAKING JUDGEMENT AND COMMUNICATION SKILLS.
TO ACCESS THE ORAL EXAM, THE GRADE OF THE WRITTEN PROOF HAS TO BE NOT LESS THAN 18/30.
THE FINAL GRADE WILL BE EXPRESSED IN THIRTIES. IT WILL BE NORMALLY THE MEAN OF PARTIAL EVALUATIONS.
THE MINIMUM GRADE (18) CORRESPONDS TO A FRAGMENTARY THEORETICAL KNOWLEDGE AND A LIMITED CAPABILITY TO USE IT IN THE APPLICATIONS.
THE MAXIMUM GRADE (30) CORRESPONDS TO A COMPLETE KNOWLEDGE OF THEORETICAL CONTENTS AND METHODOLOGIES, A CONSIDERABLE CAPABILITY TO USE IT IN THE APPLICATIONS AND COMMUNICATION SKILLS.
HONORS CAN BE OBTAINED BY A STUDENT WHO EXHIBITS A NOTEWORTHY THEORETICAL KNOWLEDGE, A PERFECT COMMAND OF SCIENTIFIC LANGUAGE AND HIGH DEGREE OF AUTONOMY ALSO IN NEW CONTEXTS.
Texts
G. ALBANO, LA PROVA SCRITTA DI GEOMETRIA: TRA TEORIA E PRATICA, MAGGIOLI EDITORE (2013).
G. ALBANO, C. D’APICE, S. SALERNO, ALGEBRA LINEARE, CUES (2002).
DIDACTICAL MATERIAL ON E-LEARNING PLATFORM.
LECTURE NOTES.