Chiara ESPOSITO | GEOMETRY I

cod. 0512300039

GEOMETRY I

	0512300039
	DEPARTMENT OF MATHEMATICS
	EQF6
	MATHEMATICS
	2024/2025

PERCORSO STANDARD EDUCATIONAL PATH

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

TEACHING UNITS

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

	Objectives
	THIS COURSE AIMS TO INTRODUCE STUDENTS TO THE THEORY OF VECTOR SPACES AND LINEAR APPLICATIONS. KNOWLEDGE AND UNDERSTANDING: THE OBJECTIVE IS TO PROVIDE STUDENTS WITH THE FUNDAMENTAL TOOLS OF LINEAR ALGEBRA AND THE THEORY OF VECTOR SPACES AND LINEAR APPLICATIONS. THIS INCLUDES UNDERSTANDING CONCEPTS SUCH AS VECTOR SPACES, SUBSPACES, LINEAR COMBINATIONS, BASES, DIMENSIONS, AND LINEAR TRANSFORMATIONS. APPLICATION OF KNOWLEDGE AND UNDERSTANDING: MOREOVER, THE COURSE AIMS TO DEVELOP STUDENTS' ABILITY TO USE THE RELATED CALCULATION TOOLS. THIS IMPLIES THE SKILL TO MANIPULATE MATRICES, SOLVE SYSTEMS OF LINEAR EQUATIONS, AND DEAL WITH ISSUES REGARDING VECTOR SPACES AND LINEAR APPLICATIONS. STUDENTS WILL BE ABLE TO APPLY THIS KNOWLEDGE TO SOLVE PRACTICAL AND THEORETICAL PROBLEMS INVOLVING VECTOR AND MATRIX MANIPULATION, SOLVING LINEAR EQUATIONS, AND UNDERSTANDING THE PROPERTIES OF LINEAR TRANSFORMATIONS. LEARNING SKILLS: THE COURSE WILL ALSO FOSTER THE DEVELOPMENT OF CRITICAL AND AUTONOMOUS LEARNING SKILLS. STUDENTS WILL BE ENCOURAGED TO DEEPEN THEIR KNOWLEDGE BEYOND THE MATERIAL PROVIDED IN CLASS THROUGH INDEPENDENT RESEARCH AND CONSULTATION OF ADDITIONAL RESOURCES. WEEKLY HOMEWORK ASSIGNMENTS WILL REQUIRE THEM TO DEMONSTRATE SMALL ASSERTIONS AUTONOMOUSLY AND EXPLORE ADVANCED OPTIONAL TOPICS. THEY WILL BE ABLE TO CRITICALLY ANALYZE NEW CONCEPTS AND THEORIES AND APPLY THEIR ACQUIRED KNOWLEDGE IN NEW AND COMPLEX CONTEXTS. COMMUNICATION SKILLS: THIS COURSE ALSO AIMS TO DEVELOP STUDENTS' COMMUNICATION SKILLS. THEY WILL BE ENCOURAGED TO CLEARLY EXPRESS COMPLEX MATHEMATICAL CONCEPTS BOTH VERBALLY AND IN WRITING, USING APPROPRIATE TECHNICAL LANGUAGE. DURING LECTURES AND EXERCISE SESSIONS, STUDENTS WILL HAVE THE OPPORTUNITY TO PRESENT AND DISCUSS THEIR PROBLEM-SOLVING SOLUTIONS, EXPLAIN THEIR REASONING, AND RESPOND TO QUESTIONS FROM THEIR PEERS AND THE INSTRUCTOR. ADDITIONALLY, THEY WILL BE ENCOURAGED TO WRITE DOWN STEPS CLARIFYING THEIR APPROACHES AND SOLUTIONS TO ASSIGNED PROBLEMS. THIS DEVELOPMENT OF COMMUNICATION SKILLS WILL CONTRIBUTE TO PREPARING STUDENTS FOR THE EFFECTIVE PRESENTATION OF THEIR IDEAS AND RESULTS. AUTONOMY OF JUDGMENT: THE COURSE ALSO AIMS TO PROMOTE STUDENTS' AUTONOMY OF JUDGMENT. THEY WILL BE ENCOURAGED TO CRITICALLY EVALUATE THE ARGUMENTS PRESENTED DURING THE COURSE. STUDENTS WILL HAVE THE OPPORTUNITY TO ANALYZE AND ASSESS DIFFERENT PROOF STRATEGIES BY COMPARING ALTERNATIVE APPROACHES. FURTHERMORE, THEY WILL BE STIMULATED TO FORMULATE QUESTIONS AND EXPLORE NEW TOPICS INDEPENDENTLY, DEMONSTRATING CRITICAL AND CREATIVE THINKING.

THIS COURSE AIMS TO INTRODUCE STUDENTS TO THE THEORY OF VECTOR SPACES AND LINEAR APPLICATIONS.

KNOWLEDGE AND UNDERSTANDING:
THE OBJECTIVE IS TO PROVIDE STUDENTS WITH THE FUNDAMENTAL TOOLS OF LINEAR ALGEBRA AND THE THEORY OF VECTOR SPACES AND LINEAR APPLICATIONS. THIS INCLUDES UNDERSTANDING CONCEPTS SUCH AS VECTOR SPACES, SUBSPACES, LINEAR COMBINATIONS, BASES, DIMENSIONS, AND LINEAR TRANSFORMATIONS.

APPLICATION OF KNOWLEDGE AND UNDERSTANDING:
MOREOVER, THE COURSE AIMS TO DEVELOP STUDENTS' ABILITY TO USE THE RELATED CALCULATION TOOLS. THIS IMPLIES THE SKILL TO MANIPULATE MATRICES, SOLVE SYSTEMS OF LINEAR EQUATIONS, AND DEAL WITH ISSUES REGARDING VECTOR SPACES AND LINEAR APPLICATIONS. STUDENTS WILL BE ABLE TO APPLY THIS KNOWLEDGE TO SOLVE PRACTICAL AND THEORETICAL PROBLEMS INVOLVING VECTOR AND MATRIX MANIPULATION, SOLVING LINEAR EQUATIONS, AND UNDERSTANDING THE PROPERTIES OF LINEAR TRANSFORMATIONS.

LEARNING SKILLS:
THE COURSE WILL ALSO FOSTER THE DEVELOPMENT OF CRITICAL AND AUTONOMOUS LEARNING SKILLS. STUDENTS WILL BE ENCOURAGED TO DEEPEN THEIR KNOWLEDGE BEYOND THE MATERIAL PROVIDED IN CLASS THROUGH INDEPENDENT RESEARCH AND CONSULTATION OF ADDITIONAL RESOURCES. WEEKLY HOMEWORK ASSIGNMENTS WILL REQUIRE THEM TO DEMONSTRATE SMALL ASSERTIONS AUTONOMOUSLY AND EXPLORE ADVANCED OPTIONAL TOPICS. THEY WILL BE ABLE TO CRITICALLY ANALYZE NEW CONCEPTS AND THEORIES AND APPLY THEIR ACQUIRED KNOWLEDGE IN NEW AND COMPLEX CONTEXTS.

COMMUNICATION SKILLS:
THIS COURSE ALSO AIMS TO DEVELOP STUDENTS' COMMUNICATION SKILLS. THEY WILL BE ENCOURAGED TO CLEARLY EXPRESS COMPLEX MATHEMATICAL CONCEPTS BOTH VERBALLY AND IN WRITING, USING APPROPRIATE TECHNICAL LANGUAGE. DURING LECTURES AND EXERCISE SESSIONS, STUDENTS WILL HAVE THE OPPORTUNITY TO PRESENT AND DISCUSS THEIR PROBLEM-SOLVING SOLUTIONS, EXPLAIN THEIR REASONING, AND RESPOND TO QUESTIONS FROM THEIR PEERS AND THE INSTRUCTOR. ADDITIONALLY, THEY WILL BE ENCOURAGED TO WRITE DOWN STEPS CLARIFYING THEIR APPROACHES AND SOLUTIONS TO ASSIGNED PROBLEMS. THIS DEVELOPMENT OF COMMUNICATION SKILLS WILL CONTRIBUTE TO PREPARING STUDENTS FOR THE EFFECTIVE PRESENTATION OF THEIR IDEAS AND RESULTS.

AUTONOMY OF JUDGMENT:
THE COURSE ALSO AIMS TO PROMOTE STUDENTS' AUTONOMY OF JUDGMENT. THEY WILL BE ENCOURAGED TO CRITICALLY EVALUATE THE ARGUMENTS PRESENTED DURING THE COURSE. STUDENTS WILL HAVE THE OPPORTUNITY TO ANALYZE AND ASSESS DIFFERENT PROOF STRATEGIES BY COMPARING ALTERNATIVE APPROACHES. FURTHERMORE, THEY WILL BE STIMULATED TO FORMULATE QUESTIONS AND EXPLORE NEW TOPICS INDEPENDENTLY, DEMONSTRATING CRITICAL AND CREATIVE THINKING.

	Prerequisites
	IT IS REQUIRED A KNOWLEDGE OF THE BASIC TOPICS IN MATHEMATICS, COVERED IN HIGH SCHOOL.

	Contents
	1. VECTOR SPACES 2. MATRICES, DETERMINANTS AND SYSTEMS OF LINEAR EQUATIONS 3. LINEAR MAPS 4. LINEAR AND BILINEAR FORMS 5. DIAGONALIZATION

	Teaching Methods
	THE TEACHING METHODS INCLUDE 64 HOURS OF FRONTAL LESSONS, DIVIDED INTO 47 HOURS OF THEORETICAL SESSIONS AND 13 HOURS OF PRACTICAL EXERCISES. KNOWLEDGE AND UNDERSTANDING: FRONTAL LESSONS ENABLE STUDENTS TO ACQUIRE THE BASIC THEORETICAL NOTIONS OF LINEAR ALGEBRA, PROVIDING THEM WITH A CLEAR UNDERSTANDING OF FUNDAMENTAL CONCEPTS. PRACTICAL EXERCISES DEMONSTRATE TO STUDENTS HOW THE THEORETICAL NOTIONS LEARNED CAN BE APPLIED TO CONCRETE EXAMPLES, HELPING THEM TO BETTER UNDERSTAND THE CONCEPTS THROUGH HANDS-ON EXPERIENCE. APPLICATION OF KNOWLEDGE AND UNDERSTANDING: THE EXERCISES PROPOSED DURING THE LESSONS SHOW STUDENTS HOW TO APPLY THEORETICAL NOTIONS IN PRACTICAL SITUATIONS, HELPING THEM DEVELOP THE ABILITY TO SOLVE PROBLEMS AUTONOMOUSLY. THE GRADUAL RESOLUTION OF EXERCISES AND PROBLEMS, BOTH IN THE CLASSROOM AND AS HOMEWORK ASSIGNMENTS, ALLOWS STUDENTS TO PROGRESSIVELY APPLY THEIR THEORETICAL KNOWLEDGE UNTIL REACHING THE FINAL RESULT, THUS CONTRIBUTING TO CONSOLIDATING THEIR UNDERSTANDING OF THE CONCEPTS. LEARNING SKILLS: THE COMBINATION OF EXERCISES SOLVED IN THE CLASSROOM AND HOMEWORK ASSIGNMENTS ENCOURAGES STUDENTS TO DEVELOP THEIR ABILITY TO SOLVE PROBLEMS AUTONOMOUSLY, PROMOTING ACTIVE AND CONTINUOUS LEARNING. STUDENTS' INVOLVEMENT IN THE GRADUAL RESOLUTION OF EXERCISES ALSO HELPS IMPROVE THEIR ABILITY TO LEARN CRITICALLY AND AUTONOMOUSLY.

Teaching Methods

THE TEACHING METHODS INCLUDE 64 HOURS OF FRONTAL LESSONS, DIVIDED INTO 47 HOURS OF THEORETICAL SESSIONS AND 13 HOURS OF PRACTICAL EXERCISES.

KNOWLEDGE AND UNDERSTANDING:
FRONTAL LESSONS ENABLE STUDENTS TO ACQUIRE THE BASIC THEORETICAL NOTIONS OF LINEAR ALGEBRA, PROVIDING THEM WITH A CLEAR UNDERSTANDING OF FUNDAMENTAL CONCEPTS. PRACTICAL EXERCISES DEMONSTRATE TO STUDENTS HOW THE THEORETICAL NOTIONS LEARNED CAN BE APPLIED TO CONCRETE EXAMPLES, HELPING THEM TO BETTER UNDERSTAND THE CONCEPTS THROUGH HANDS-ON EXPERIENCE.

APPLICATION OF KNOWLEDGE AND UNDERSTANDING:
THE EXERCISES PROPOSED DURING THE LESSONS SHOW STUDENTS HOW TO APPLY THEORETICAL NOTIONS IN PRACTICAL SITUATIONS, HELPING THEM DEVELOP THE ABILITY TO SOLVE PROBLEMS AUTONOMOUSLY. THE GRADUAL RESOLUTION OF EXERCISES AND PROBLEMS, BOTH IN THE CLASSROOM AND AS HOMEWORK ASSIGNMENTS, ALLOWS STUDENTS TO PROGRESSIVELY APPLY THEIR THEORETICAL KNOWLEDGE UNTIL REACHING THE FINAL RESULT, THUS CONTRIBUTING TO CONSOLIDATING THEIR UNDERSTANDING OF THE CONCEPTS.

LEARNING SKILLS:
THE COMBINATION OF EXERCISES SOLVED IN THE CLASSROOM AND HOMEWORK ASSIGNMENTS ENCOURAGES STUDENTS TO DEVELOP THEIR ABILITY TO SOLVE PROBLEMS AUTONOMOUSLY, PROMOTING ACTIVE AND CONTINUOUS LEARNING. STUDENTS' INVOLVEMENT IN THE GRADUAL RESOLUTION OF EXERCISES ALSO HELPS IMPROVE THEIR ABILITY TO LEARN CRITICALLY AND AUTONOMOUSLY.

	Verification of learning
	LA SEZIONE MODALITÀ DI VERIFICA AGGIORNATA, IN ITALIANO E IN INGLESE È QUESTA: LA STRUTTURA DELL'ESAME MIRA A VALUTARE SIA LA CAPACITÀ DEGLI STUDENTI DI RISOLVERE PROBLEMI PRATICI CHE LA LORO COMPRENSIONE TEORICA DEI CONCETTI DISCUSSI DURANTE IL CORSO. L’ESAME CONSISTE IN UNA PROVA SCRITTA ED UNA PROVA ORALE. NELLA PROVA SCRITTA DELLA DURATA DI 2 ORE E 30 LO STUDENTE SARÀ SOTTOPOSTO ALLA RISOLUZIONE DI ESERCIZI FINALIZZATI A VERIFICARE LA CAPACITÀ DELLO STUDENTE DI APPLICARE LE METODOLOGIE DELL’ALGEBRA LINEARE E DEGLI SPAZI VETTORIALI. LA PROVA ORALE DELLA DURATA DI CIRCA 30 MINUTI È FINALIZZATA A VERIFICARE LE CONOSCENZE TEORICHE DELLO STUDENTE E LA CAPACITÀ DI SVILUPPARE IN AUTONOMIA LE ARGOMENTAZIONI PRESENTATE DURANTE IL CORSO. LA PROVA SCRITTA RICHIEDE L’APPLICAZIONE DELLE METODOLOGIE DELL’ALGEBRA LINEARE E DEGLI SPAZI VETTORIALI, QUALI LA SOLUZIONE DEI SISTEMI DI EQUAZIONI LINEARI, IL CALCOLO MATRICIALE, LO STUDIO DEL NUCLEO E DELL’IMMAGINE DI UN’APPLICAZIONE LINEARE E LE RAPPRESENTAZIONI DEI SOTTOSPAZI VETTORIALI. LA PROVA ORALE SARÀ ACCESSIBILE AGLI STUDENTI CHE SUPERANO LA PROVA SCRITTA. . IL PUNTEGGIO DELLA PROVA SCRITTA NON DETERMINA IL VOTO FINALE. IL PUNTEGGIO DELLA PROVA ORALE DIPENDE DAL GRADO DI APPROFONDIMENTO E DALLA CAPACITÀ DELLO STUDENTE DI PRESENTARE EFFICACEMENTE I CONTENUTI E DISCUTERE CRITICAMENTE GLI ARGOMENTI ILLUSTRATI DURANTE IL CORSO. CONOSCENZA E COMPRENSIONE: LA PARTE SCRITTA DELL'ESAME, BASATA SU ESERCIZI SIMILI A QUELLI ASSEGNATI COME COMPITI DURANTE L'ANNO, PERMETTE AGLI STUDENTI DI DIMOSTRARE LA LORO COMPRENSIONE PRATICA DEI CONCETTI DI ALGEBRA LINEARE LA PARTE ORALE, CHE VERTE SU DOMANDE DI TEORIA SULLE DEFINIZIONI, I TEOREMI E LE DIMOSTRAZIONI DISCUSSE DURANTE IL CORSO, VALUTA LA PROFONDITÀ DELLA COMPRENSIONE TEORICA DEGLI STUDENTI. CAPACITÀ DI APPLICARE CONOSCENZA E COMPRENSIONE: LA PROVA SCRITTA PERMETTE AGLI STUDENTI DI DIMOSTRARE LA LORO CAPACITÀ DI APPLICARE LE CONOSCENZE TEORICHE ACQUISITE DURANTE IL CORSO ALLA RISOLUZIONE DI PROBLEMI PRATICI, SIMILI A QUELLI AFFRONTATI DURANTE LE LEZIONI E I COMPITI A CASA. DURANTE L'ESAME ORALE, GLI STUDENTI SONO CHIAMATI A RISPONDERE A DOMANDE DI TEORIA CHE RICHIEDONO LORO DI APPLICARE LE NOZIONI ACQUISITE PER SPIEGARE CONCETTI, DEFINIZIONI E DIMOSTRAZIONI. CAPACITÀ DI APPRENDERE: LA COMBINAZIONE DI PROVA SCRITTA E ORALE PERMETTE AGLI STUDENTI DI DIMOSTRARE LA LORO CAPACITÀ DI APPRENDERE IN MODO CRITICO E AUTONOMO, SIA NELLA RISOLUZIONE DI PROBLEMI PRATICI CHE NELLA COMPRENSIONE TEORICA DEI CONCETTI. CAPACITÀ DI COMUNICARE: DURANTE L'ESAME ORALE, GLI STUDENTI DEVONO ESSERE IN GRADO DI COMUNICARE IN MODO CHIARO LE LORO CONOSCENZE E COMPRENSIONI TEORICHE ACQUISITE DURANTE IL CORSO. IN PARTICOLARE, DEVONO ESSERE IN GRADO DI SPIEGARE CONCETTI, DEFINIZIONI, TEOREMI E DIMOSTRAZIONI IN MODO COERENTE E COMPRENSIBILE. IL VOTO FINALE, ESPRESSO IN TRENTESIMI, È DETERMINATO PRINCIPALMENTE DAL COLLOQUIO ORALE. IL PUNTEGGIO MINIMO PREVEDE LA CONOSCENZA DEI SEGUENTI TEMI: SPAZI VETTORIALI APPLICAZIONI LINEARI, FORME LINEARI, ALGEBRA LINEARE, DIAGONALIZZAZIONE DEGLI ENDOMORFISMI. LA LODE VIENE ASSEGNATA AGLI STUDENTI CHE DIMOSTRANO UNA PIENA PADRONANZA DELL'ARGOMENTO E UNA CORRETTA FORMALIZZAZIONE DELLE CONOSCENZE DURANTE IL COLLOQUIO. THE STRUCTURE OF THE EXAM AIMS TO ASSESS BOTH STUDENTS' ABILITY TO SOLVE PRACTICAL PROBLEMS AND THEIR THEORETICAL UNDERSTANDING OF THE CONCEPTS DISCUSSED DURING THE COURSE. THE EXAM CONSISTS OF A WRITTEN TEST AND AN ORAL TEST. IN THE WRITTEN TEST LASTING 2 HOURS AND 30 MINUTES THE STUDENT WILL BE SUBJECT TO THE SOLUTION OF EXERCISES AIMED AT VERIFYING THE STUDENT'S ABILITY TO APPLY THE METHODOLOGIES OF LINEAR ALGEBRA AND VECTOR SPACE. THE ORAL EXAM LASTING ABOUT 30 MINUTES IS AIMED AT VERIFYING THE STUDENT'S THEORETICAL KNOWLEDGE AND THE ABILITY TO DEVELOP THE ARGUMENTS PRESENTED DURING THE COURSE INDEPENDENTLY. THE WRITTEN TEST REQUIRES THE APPLICATION OF THE METHODOLOGIES OF LINEAR ALGEBRA AND VECTOR SPACES, SUCH AS THE SOLUTION OF SYSTEMS OF LINEAR EQUATIONS, MATRICIAL CALCULUS, THE STUDY OF THE NUCLEUS AND THE IMAGE OF A LINEAR APPLICATION AND THE REPRESENTATIONS OF VECTOR SUBSPACE . THE ORAL TEST WILL BE ACCESSIBLE TO STUDENTS WHO PASS THE WRITTEN TEST. THE SCORE OF THE WRITTEN TEST DOES NOT DETERMINE THE FINAL GRADE. THE SCORE OF THE ORAL EXAM DEPENDS ON THE DEGREE OF DEPTH AND THE ABILITY OF THE STUDENT TO EFFECTIVELY PRESENT THE CONTENTS AND CRITICALLY DISCUSS THE TOPICS ILLUSTRATED DURING THE COURSE. KNOWLEDGE AND UNDERSTANDING: THE WRITTEN PORTION OF THE EXAM, BASED ON EXERCISES SIMILAR TO THOSE ASSIGNED AS HOMEWORK DURING THE YEAR, ALLOWS STUDENTS TO DEMONSTRATE THEIR PRACTICAL UNDERSTANDING OF LINEAR ALGEBRA CONCEPTS THE ORAL PART, WHICH FOCUSES ON THEORY QUESTIONS ON THE DEFINITIONS, THEOREMS AND PROOFS DISCUSSED DURING THE COURSE, ASSESSES THE DEPTH OF THE STUDENTS' THEORETICAL UNDERSTANDING. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: THE WRITTEN TEST ALLOWS STUDENTS TO DEMONSTRATE THEIR ABILITY TO APPLY THE THEORETICAL KNOWLEDGE ACQUIRED DURING THE COURSE TO THE RESOLUTION OF PRACTICAL PROBLEMS, SIMILAR TO THOSE ADDRESSED DURING LESSONS AND HOMEWORK. DURING THE ORAL EXAM, STUDENTS ARE CALLED TO ANSWER THEORY QUESTIONS WHICH REQUIRE THEM TO APPLY THE NOTIONS ACQUIRED TO EXPLAIN CONCEPTS, DEFINITIONS AND DEMONSTRATIONS. LEARNING ABILITY: THE COMBINATION OF WRITTEN AND ORAL TEST ALLOWS STUDENTS TO DEMONSTRATE THEIR ABILITY TO LEARN IN A CRITICAL AND INDEPENDENT WAY, BOTH IN THE SOLVING OF PRACTICAL PROBLEMS AND IN THEORETICAL UNDERSTANDING OF CONCEPTS. ABILITY TO COMMUNICATE: DURING THE ORAL EXAM, STUDENTS MUST BE ABLE TO CLEARLY COMMUNICATE THEIR THEORETICAL KNOWLEDGE AND UNDERSTANDINGS ACQUIRED DURING THE COURSE. IN PARTICULAR, THEY MUST BE ABLE TO EXPLAIN CONCEPTS, DEFINITIONS, THEOREMS AND PROOFS IN A COHERENT AND UNDERSTANDABLE WAY. THE FINAL MARK, EXPRESSED IN THIRTIETH, IS MAINLY DETERMINED BY THE ORAL INTERVIEW. THE MINIMUM SCORE REQUIRES KNOWLEDGE OF THE FOLLOWING TOPICS: VECTOR SPACES LINEAR APPLICATIONS, LINEAR FORMS, LINEAR ALGEBRA, DIAGONALISATION OF ENDOMORPHISM. LAUDE IS AWARDED TO STUDENTS WHO DEMONSTRATE FULL MASTERY OF THE TOPIC AND CORRECT FORMALIZATION OF KNOWLEDGE DURING THE INTERVIEW.

Verification of learning

LA SEZIONE MODALITÀ DI VERIFICA AGGIORNATA, IN ITALIANO E IN INGLESE È QUESTA:

LA STRUTTURA DELL'ESAME MIRA A VALUTARE SIA LA CAPACITÀ DEGLI STUDENTI DI RISOLVERE PROBLEMI PRATICI CHE LA LORO COMPRENSIONE TEORICA DEI CONCETTI DISCUSSI DURANTE IL CORSO.

L’ESAME CONSISTE IN UNA PROVA SCRITTA ED UNA PROVA ORALE. NELLA PROVA SCRITTA DELLA DURATA DI 2 ORE E 30 LO STUDENTE SARÀ SOTTOPOSTO ALLA RISOLUZIONE DI ESERCIZI FINALIZZATI A VERIFICARE LA CAPACITÀ DELLO STUDENTE DI APPLICARE LE METODOLOGIE DELL’ALGEBRA LINEARE E DEGLI SPAZI VETTORIALI. LA PROVA ORALE DELLA DURATA DI CIRCA 30 MINUTI È FINALIZZATA A VERIFICARE LE CONOSCENZE TEORICHE DELLO STUDENTE E LA CAPACITÀ DI SVILUPPARE IN AUTONOMIA LE ARGOMENTAZIONI PRESENTATE DURANTE IL CORSO.

LA PROVA SCRITTA RICHIEDE L’APPLICAZIONE DELLE METODOLOGIE DELL’ALGEBRA LINEARE E DEGLI SPAZI VETTORIALI, QUALI LA SOLUZIONE DEI SISTEMI DI EQUAZIONI LINEARI, IL CALCOLO MATRICIALE, LO STUDIO DEL NUCLEO E DELL’IMMAGINE DI UN’APPLICAZIONE LINEARE E LE RAPPRESENTAZIONI DEI SOTTOSPAZI VETTORIALI.

LA PROVA ORALE SARÀ ACCESSIBILE AGLI STUDENTI CHE SUPERANO LA PROVA SCRITTA. . IL PUNTEGGIO DELLA PROVA SCRITTA NON DETERMINA IL VOTO FINALE. IL PUNTEGGIO DELLA PROVA ORALE DIPENDE DAL GRADO DI APPROFONDIMENTO E DALLA CAPACITÀ DELLO STUDENTE DI PRESENTARE EFFICACEMENTE I CONTENUTI E DISCUTERE CRITICAMENTE GLI ARGOMENTI ILLUSTRATI DURANTE IL CORSO.

CONOSCENZA E COMPRENSIONE:
LA PARTE SCRITTA DELL'ESAME, BASATA SU ESERCIZI SIMILI A QUELLI ASSEGNATI COME COMPITI DURANTE L'ANNO, PERMETTE AGLI STUDENTI DI DIMOSTRARE LA LORO COMPRENSIONE PRATICA DEI CONCETTI DI ALGEBRA LINEARE
LA PARTE ORALE, CHE VERTE SU DOMANDE DI TEORIA SULLE DEFINIZIONI, I TEOREMI E LE DIMOSTRAZIONI DISCUSSE DURANTE IL CORSO, VALUTA LA PROFONDITÀ DELLA COMPRENSIONE TEORICA DEGLI STUDENTI.

CAPACITÀ DI APPLICARE CONOSCENZA E COMPRENSIONE:
LA PROVA SCRITTA PERMETTE AGLI STUDENTI DI DIMOSTRARE LA LORO CAPACITÀ DI APPLICARE LE CONOSCENZE TEORICHE ACQUISITE DURANTE IL CORSO ALLA RISOLUZIONE DI PROBLEMI PRATICI, SIMILI A QUELLI AFFRONTATI DURANTE LE LEZIONI E I COMPITI A CASA.
DURANTE L'ESAME ORALE, GLI STUDENTI SONO CHIAMATI A RISPONDERE A DOMANDE DI TEORIA CHE RICHIEDONO LORO DI APPLICARE LE NOZIONI ACQUISITE PER SPIEGARE CONCETTI, DEFINIZIONI E DIMOSTRAZIONI.

CAPACITÀ DI APPRENDERE:
LA COMBINAZIONE DI PROVA SCRITTA E ORALE PERMETTE AGLI STUDENTI DI DIMOSTRARE LA LORO CAPACITÀ DI APPRENDERE IN MODO CRITICO E AUTONOMO, SIA NELLA RISOLUZIONE DI PROBLEMI PRATICI CHE NELLA COMPRENSIONE TEORICA DEI CONCETTI.

CAPACITÀ DI COMUNICARE:

DURANTE L'ESAME ORALE, GLI STUDENTI DEVONO ESSERE IN GRADO DI COMUNICARE IN MODO CHIARO LE LORO CONOSCENZE E COMPRENSIONI TEORICHE ACQUISITE DURANTE IL CORSO. IN PARTICOLARE, DEVONO ESSERE IN GRADO DI SPIEGARE CONCETTI, DEFINIZIONI, TEOREMI E DIMOSTRAZIONI IN MODO COERENTE E COMPRENSIBILE.

IL VOTO FINALE, ESPRESSO IN TRENTESIMI, È DETERMINATO PRINCIPALMENTE DAL COLLOQUIO ORALE. IL PUNTEGGIO MINIMO PREVEDE LA CONOSCENZA DEI SEGUENTI TEMI: SPAZI VETTORIALI APPLICAZIONI LINEARI, FORME LINEARI, ALGEBRA LINEARE, DIAGONALIZZAZIONE DEGLI ENDOMORFISMI. LA LODE VIENE ASSEGNATA AGLI STUDENTI CHE DIMOSTRANO UNA PIENA PADRONANZA DELL'ARGOMENTO E UNA CORRETTA FORMALIZZAZIONE DELLE CONOSCENZE DURANTE IL COLLOQUIO.

THE STRUCTURE OF THE EXAM AIMS TO ASSESS BOTH STUDENTS' ABILITY TO SOLVE PRACTICAL PROBLEMS AND THEIR THEORETICAL UNDERSTANDING OF THE CONCEPTS DISCUSSED DURING THE COURSE.
THE EXAM CONSISTS OF A WRITTEN TEST AND AN ORAL TEST. IN THE WRITTEN TEST LASTING 2 HOURS AND 30 MINUTES THE STUDENT WILL BE SUBJECT TO THE SOLUTION OF EXERCISES AIMED AT VERIFYING THE STUDENT'S ABILITY TO APPLY THE METHODOLOGIES OF LINEAR ALGEBRA AND VECTOR SPACE. THE ORAL EXAM LASTING ABOUT 30 MINUTES IS AIMED AT VERIFYING THE STUDENT'S THEORETICAL KNOWLEDGE AND THE ABILITY TO DEVELOP THE ARGUMENTS PRESENTED DURING THE COURSE INDEPENDENTLY.
THE WRITTEN TEST REQUIRES THE APPLICATION OF THE METHODOLOGIES OF LINEAR ALGEBRA AND VECTOR SPACES, SUCH AS THE SOLUTION OF SYSTEMS OF LINEAR EQUATIONS, MATRICIAL CALCULUS, THE STUDY OF THE NUCLEUS AND THE IMAGE OF A LINEAR APPLICATION AND THE REPRESENTATIONS OF VECTOR SUBSPACE .
THE ORAL TEST WILL BE ACCESSIBLE TO STUDENTS WHO PASS THE WRITTEN TEST. THE SCORE OF THE WRITTEN TEST DOES NOT DETERMINE THE FINAL GRADE. THE SCORE OF THE ORAL EXAM DEPENDS ON THE DEGREE OF DEPTH AND THE ABILITY OF THE STUDENT TO EFFECTIVELY PRESENT THE CONTENTS AND CRITICALLY DISCUSS THE TOPICS ILLUSTRATED DURING THE COURSE.
KNOWLEDGE AND UNDERSTANDING:
THE WRITTEN PORTION OF THE EXAM, BASED ON EXERCISES SIMILAR TO THOSE ASSIGNED AS HOMEWORK DURING THE YEAR, ALLOWS STUDENTS TO DEMONSTRATE THEIR PRACTICAL UNDERSTANDING OF LINEAR ALGEBRA CONCEPTS THE ORAL PART, WHICH FOCUSES ON THEORY QUESTIONS ON THE DEFINITIONS, THEOREMS AND PROOFS DISCUSSED DURING THE COURSE, ASSESSES THE DEPTH OF THE STUDENTS' THEORETICAL UNDERSTANDING.
ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
THE WRITTEN TEST ALLOWS STUDENTS TO DEMONSTRATE THEIR ABILITY TO APPLY THE THEORETICAL KNOWLEDGE ACQUIRED DURING THE COURSE TO THE RESOLUTION OF PRACTICAL PROBLEMS, SIMILAR TO THOSE ADDRESSED DURING LESSONS AND HOMEWORK. DURING THE ORAL EXAM, STUDENTS ARE CALLED TO ANSWER THEORY QUESTIONS WHICH REQUIRE THEM TO APPLY THE NOTIONS ACQUIRED TO EXPLAIN CONCEPTS, DEFINITIONS AND DEMONSTRATIONS.
LEARNING ABILITY:
THE COMBINATION OF WRITTEN AND ORAL TEST ALLOWS STUDENTS TO DEMONSTRATE THEIR ABILITY TO LEARN IN A CRITICAL AND INDEPENDENT WAY, BOTH IN THE SOLVING OF PRACTICAL PROBLEMS AND IN THEORETICAL UNDERSTANDING OF CONCEPTS.
ABILITY TO COMMUNICATE:
DURING THE ORAL EXAM, STUDENTS MUST BE ABLE TO CLEARLY COMMUNICATE THEIR THEORETICAL KNOWLEDGE AND UNDERSTANDINGS ACQUIRED DURING THE COURSE. IN PARTICULAR, THEY MUST BE ABLE TO EXPLAIN CONCEPTS, DEFINITIONS, THEOREMS AND PROOFS IN A COHERENT AND UNDERSTANDABLE WAY. THE FINAL MARK, EXPRESSED IN THIRTIETH, IS MAINLY DETERMINED BY THE ORAL INTERVIEW. THE MINIMUM SCORE REQUIRES KNOWLEDGE OF THE FOLLOWING TOPICS: VECTOR SPACES LINEAR APPLICATIONS, LINEAR FORMS, LINEAR ALGEBRA, DIAGONALISATION OF ENDOMORPHISM. LAUDE IS AWARDED TO STUDENTS WHO DEMONSTRATE FULL MASTERY OF THE TOPIC AND CORRECT FORMALIZATION OF KNOWLEDGE DURING THE INTERVIEW.

	Texts
	LECTURE NOTES E. SERNESI, GEOMETRIA 1, BOLLATI BORINGHIERI. S. LIPSCHUTZ, ALGEBRA LINEARE MCGRAW-HILL.

	More Information
	CHESPOSITO@UNISA.IT

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Chiara ESPOSITO | GEOMETRY I

GEOMETRY I

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Chiara ESPOSITO | GEOMETRY I