Ingegneria Edile-Architettura | GEOMETRIA

cod. 0660100002

GEOMETRIA

	0660100002
	DIPARTIMENTO DI INGEGNERIA CIVILE
	CORSO DI LAUREA MAGISTRALE A CICLO UNICO DI 5 ANNI
	BUILDING ENGINEERING - ARCHITECTURE
	2013/2014

PERCORSO COMUNE

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2012
	PRIMO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/03	6	60	LESSONS	BASIC COMPULSORY SUBJECTS

	Objectives
	THE COURSE AIMS AT THE ACQUISITION OF THE BASIC ELEMENTS OF LINEAR ALGEBRA AND ANALYTIC GEOMETRY: MATRICES, LINEAR SYSTEMS, VECTOR SPACES, EUCLIDEAN SPACES, LINEAR OPERATORS, EIGENSPACES, DIAGONALIZATION, 2D AND 3D ANALYTICAL GEOMETRY. COURSE’S LEARNING OUTCOMES CONSIST IN THE ACHIEVEMENT OF RESULTS AND PROOF TECHNIQUES, AS WELL AS THE ABILITY TO USE THE RELATED COMPUTATIONAL TOOLS. THE COURSE'S MAIN AIM IS TO STRENGTHEN BASIC MATHEMATICAL KNOWLEDGE AND TO PROVIDE AND DEVELOP TOOLS USEFUL FOR A SCIENTIFIC APPROACH TO PROBLEMS AND PHENOMENA THAT STUDENTS ARE GOING TO ENCOUNTER DURING THE REST OF THEIR STUDIES. THE THEORETICAL PART OF THE COURSE WILL BE PRESENTED IN A RIGOROUS AND CONCISE WAY AND IT WILL BE SUPPORTED BY PARALLEL EXERCISE SESSIONS DESIGNED TO PROMOTE MEANINGFUL UNDERSTANDING OF CONCEPTS.

THE COURSE AIMS AT THE ACQUISITION OF THE BASIC ELEMENTS OF LINEAR ALGEBRA AND ANALYTIC GEOMETRY: MATRICES, LINEAR SYSTEMS, VECTOR SPACES, EUCLIDEAN SPACES, LINEAR OPERATORS, EIGENSPACES, DIAGONALIZATION, 2D AND 3D ANALYTICAL GEOMETRY. COURSE’S LEARNING OUTCOMES CONSIST IN THE ACHIEVEMENT OF RESULTS AND PROOF TECHNIQUES, AS WELL AS THE ABILITY TO USE THE RELATED COMPUTATIONAL TOOLS. THE COURSE'S MAIN AIM IS TO STRENGTHEN BASIC MATHEMATICAL KNOWLEDGE AND TO PROVIDE AND DEVELOP TOOLS USEFUL FOR A SCIENTIFIC APPROACH TO PROBLEMS AND PHENOMENA THAT STUDENTS ARE GOING TO ENCOUNTER DURING THE REST OF THEIR STUDIES. THE THEORETICAL PART OF THE COURSE WILL BE PRESENTED IN A RIGOROUS AND CONCISE WAY AND IT WILL BE SUPPORTED BY PARALLEL EXERCISE SESSIONS DESIGNED TO PROMOTE MEANINGFUL UNDERSTANDING OF CONCEPTS.

	Prerequisites
	FOR THE SUCCESSFUL ACHIEVEMENT OF THE GOALS, STUDENTS ARE REQUIRED TO MASTER KNOWLEDGE ABOUT BASIC MATHEMATICS AS PREREQUISITES.

	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 4,PRACTICE HRS 3). LINEAR SYSTEMS OF EQUATIONS: DEFINITIONS, MATRIX EQUATION, CONSISTENCY AND INCONSISTENCY, NUMBER OF SOLUTIONS. ROUCHÉ-CAPELLI THEOREM. CRAMER RULE. GAUSSIAN ELIMINATION RULE. BASIS OF THE SOLUTIONS SET OF A HOMOGENEOUS SYSTEM. DISCUSSION OF LINEAR SYSTEMS WITH A PARAMETER. (LECTURES HRS 4,PRACTICE HRS 4). THE VECTOR SPACE STRUCTURE. LINEAR DEPENDENCE AND INDEPENDENCE OF VECTORS. GENERATORS. BASES. STEINITZ'S LEMMA. DIMENSION OF A VECTOR SPACE AND RELATED THEOREM. VECTOR SUBSPACES. INTERSECTION AND SUM, 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. COMPONENTS IN AN ORTHONORMAL BASIS. ORTHOGONAL PROJECTIONS. GRAM-SCHMIDT PROCEDURE. (LECTURES HRS 4,PRACTICE HRS 3). LINEAR OPERATORS. DEFINITIONS OF LINEAR OPERATOR (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. ALGEBARIC AND GEOMETRIC MULTEPLICITIES.DEFINITION OF DIAGONALIZATION AND ITS CHARACTERIZATIONS (FOR MATRICES AND ENDOMORPHISMS). MAIN THEOREM. SUFFICIENT CONDITION FOR THE DIAGONALIZATION. ORTHOGONAL 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 (ALGEBRAIC, 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 CANOLICAL FORMS. ALGORITHM FOR REDUCING CONICS TO CANONICAL FORM. CHARACTERISTIC GEOMETRIC PARAMETERS. (LECTURES HRS 4,PRACTICE HRS 3). 3D ANALITYC GEOMETRY. SPATIAL CARTESIAN COORDINATE SYSTEM. EQUATION OF THE PLANE (CARTESIAN AND PARAMETRIC). EQUATION OF THE LINE (PARAMETRIC, CARTESIAN, SYMMETRIC). BUNDLES OF PLANS. STARS OF PLANS. CONDITIONS FOR PARALLEL AND PERPENDICULAR LINES, LINE AND PLAN, PLANS.NON-COPLANAR LINES. ANGLE BETWEEN TWO PLANS, TWO LINES, A LINE AND A PLANE. DISTANCES AND SYMMETRIES IN THE SPACE. PROBLEMS ON PARALLELISM AND PERPENDICULARITY IN THE PLANE. QUADRIICS: CLASSIFICATION AND CANONICAL FORMS. ALGORITHM FOR REDUCING CONICS TO CANONICAL FORM. CHARACTERISTIC GEOMETRIC PARAMETERS. (LECTURES HRS 5,PRACTICE HRS 4).

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 4,PRACTICE HRS 3).
LINEAR SYSTEMS OF EQUATIONS: DEFINITIONS, MATRIX EQUATION, CONSISTENCY AND INCONSISTENCY, NUMBER OF SOLUTIONS. ROUCHÉ-CAPELLI THEOREM. CRAMER RULE. GAUSSIAN ELIMINATION RULE. BASIS OF THE SOLUTIONS SET OF A HOMOGENEOUS SYSTEM. DISCUSSION OF LINEAR SYSTEMS WITH A PARAMETER. (LECTURES HRS 4,PRACTICE HRS 4).
THE VECTOR SPACE STRUCTURE. LINEAR DEPENDENCE AND INDEPENDENCE OF VECTORS. GENERATORS. BASES. STEINITZ'S LEMMA. DIMENSION OF A VECTOR SPACE AND RELATED THEOREM. VECTOR SUBSPACES. INTERSECTION AND SUM, 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. COMPONENTS IN AN ORTHONORMAL BASIS. ORTHOGONAL PROJECTIONS. GRAM-SCHMIDT PROCEDURE. (LECTURES HRS 4,PRACTICE HRS 3).
LINEAR OPERATORS. DEFINITIONS OF LINEAR OPERATOR (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. ALGEBARIC AND GEOMETRIC MULTEPLICITIES.DEFINITION OF DIAGONALIZATION AND ITS CHARACTERIZATIONS (FOR MATRICES AND ENDOMORPHISMS). MAIN THEOREM. SUFFICIENT CONDITION FOR THE DIAGONALIZATION. ORTHOGONAL 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 (ALGEBRAIC, 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 CANOLICAL FORMS. ALGORITHM FOR REDUCING CONICS TO CANONICAL FORM. CHARACTERISTIC GEOMETRIC PARAMETERS. (LECTURES HRS 4,PRACTICE HRS 3).
3D ANALITYC GEOMETRY. SPATIAL CARTESIAN COORDINATE SYSTEM. EQUATION OF THE PLANE (CARTESIAN AND PARAMETRIC). EQUATION OF THE LINE (PARAMETRIC, CARTESIAN, SYMMETRIC). BUNDLES OF PLANS. STARS OF PLANS. CONDITIONS FOR PARALLEL AND PERPENDICULAR LINES, LINE AND PLAN, PLANS.NON-COPLANAR LINES. ANGLE BETWEEN TWO PLANS, TWO LINES, A LINE AND A PLANE. DISTANCES AND SYMMETRIES IN THE SPACE. PROBLEMS ON PARALLELISM AND PERPENDICULARITY IN THE PLANE. QUADRIICS: CLASSIFICATION AND CANONICAL FORMS. ALGORITHM FOR REDUCING CONICS TO CANONICAL FORM. CHARACTERISTIC GEOMETRIC PARAMETERS. (LECTURES HRS 5,PRACTICE HRS 4).

	Teaching Methods
	THE COURSE COVERS THEORETICAL LECTURES, DEVOTED TO THE FACE-TO-FACE DELIVERY OF ALL THE COURSE CONTENTS, AND CLASSROOM PRACTICE DEVOTED TO PROVIDE THE STUDENTS WITH THE MAIN TOOLS NEEDED TO PROBLEM-SOLVING ACTIVITIES RELATED TO THE COURSE CONTENTS.

	Verification of learning
	THE ASSESSMENT OF THE EXPECTED LEARNING OUTCOMES WILL BE CARRIED OUT BY MEANS OF A WRITTEN TEST AND AN ORAL INTERVIEW. TO PASS THE EXAM THE STUDENT MUST PROVE TO HAVE UNDERSTOOD AND TO BE ABLE TO MASTER THE MAIN CONCEPTS OF THE COURSE TOPICS. THE FINAL MARK, OUT OF THIRTY (EVENTUALLY CUM LAUDE), WILL DEPEND ON THE MASTERING ABILITY OF THE COURSE CONTENTS, TAKING INTO ACCOUNT THE QUALITY OF THE WRITTEN AND ORAL ELABORATION AND THE SELF-EVALUATION CAPABILITY SHOWN.

Verification of learning

THE ASSESSMENT OF THE EXPECTED LEARNING OUTCOMES WILL BE CARRIED OUT BY MEANS OF A WRITTEN TEST AND AN ORAL INTERVIEW. TO PASS THE EXAM THE STUDENT MUST PROVE TO HAVE UNDERSTOOD AND TO BE ABLE TO MASTER THE MAIN CONCEPTS OF THE COURSE TOPICS. THE FINAL MARK, OUT OF THIRTY (EVENTUALLY CUM LAUDE), WILL DEPEND ON THE MASTERING ABILITY OF THE COURSE CONTENTS, TAKING INTO ACCOUNT THE QUALITY OF THE WRITTEN AND ORAL ELABORATION AND THE SELF-EVALUATION CAPABILITY SHOWN.

	Texts
	G. ALBANO, LA PROVA SCRITTA DI GEOMETRIA: TRA TEORIA E PRATICA, CUES (2011). G. ALBANO, C. D’APICE, S. SALERNO, ALGEBRA LINEARE, CUES (2002). EDUCATIONAL CONTENTS ON E-LEARNING PLATFORM IWT. LECTURE NOTES.

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