Chiara ESPOSITO | GEOMETRY I

cod. MT12300051

GEOMETRY I

	MT12300051
	DEPARTMENT OF MATHEMATICS
	EQF6
	MATHEMATICS
	2025/2026

PERCORSO STANDARD EDUCATIONAL PATH

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

TEACHING UNITS

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

PROFESSORS

	CHIARA ESPOSITO T Curriculum
	GIOVANNI SPARANO Curriculum

	Objectives
	OBIETTIVO GENERALE THIS COURSE AIMS TO INTRODUCE STUDENTS TO THE THEORY OF VECTOR SPACES AND LINEAR APPLICATIONS. CONOSCENZA E CAPACITÀ DI COMPRENSIONE 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. CAPACITÀ DI APPLICARE CONOSCENZA E COMPRENSIONE 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. AUTONOMIA DI GIUDIZIO 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. ABILITÀ COMUNICATIVE 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. CAPACITÀ DI APPRENDIMENTO 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.

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

CONOSCENZA E CAPACITÀ DI COMPRENSIONE
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.

CAPACITÀ DI APPLICARE CONOSCENZA E COMPRENSIONE
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.

AUTONOMIA DI GIUDIZIO
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.

ABILITÀ COMUNICATIVE
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.

CAPACITÀ DI APPRENDIMENTO
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.

	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 FORMS 5. DIAGONALIZATION

	Teaching Methods
	THE COURSE CONSISTS IN 64 FRONT TEACHING HOURS, DIVIDED INTO 50 HOURS OF THEORETICAL LECTURES AND 14 HOURS OF EXERCISES. DURING THE LECTURES WE WILL DISCUSS THEORETIC TOPICS, TOGETHER WITH CONCRETE PROBLEMS TO BE SOLVED BY THE STUDENTS UNDER THE INSTRUCTOR GUIDANCE, DURING THE EXERCISE SESSIONS

	Verification of learning
	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

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 R. ESPOSITO, A. RUSSO, LEZIONI DI GEOMETRIA, PARTE PRIMA, LIGUORI. E. SERNESI, GEOMETRIA 1, BOLLATI BORINGHIERI. M. MANETTI, ALGEBRA LINEARE PER MATEMATICI. S. LIPSCHUTZ, ALGEBRA LINEARE MCGRAW-HILL.

	More Information
	EMAIL: CHESPOSITO@UNISA.IT

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Chiara ESPOSITO | GEOMETRY I