Giovannina ALBANO | GEOMETRY, ALGEBRA AND LOGIC

cod. 0612700114

GEOMETRY, ALGEBRA AND LOGIC

	0612700114
	DEPARTMENT OF INFORMATION AND ELECTRICAL ENGINEERING AND APPLIED MATHEMATICS
	EQF6
	COMPUTER ENGINEERING
	2024/2025

CURRICULUM NETWORKS

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2022
	SPRING SEMESTER

TEACHING UNITS

SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
MAT/03	5	40	LESSONS	BASIC COMPULSORY SUBJECTS
MAT/03	4	32	EXERCISES	BASIC COMPULSORY SUBJECTS

	Objectives
	THE COURSE PROPOSES THE BASIC ELEMENTS OF LINEAR ALGEBRA, GEOMETRY AND MATHEMATICAL LOGIC. THE EDUCATIONAL OBJECTIVES OF THE COURSE ARE THE ACHIEVEMENT OF THE RESULTS AND PROOF TECHNIQUES, AS WELL AS THE ABILITY TO USE THE RELEVANT COMPUTATIONAL TOOLS. KNOWLEDGE AND UNDERSTANDING CAPABILITIES. MATRICES AND LINEAR SYSTEMS. VECTOR AND EUCLIDEAN SPACES. 2D AND 3D ANALYTIC GEOMETRY. PROPOSITIONAL AND PREDICATIVE LOGIC. SOFTWARE TOOLS FOR LINEAR ALGEBRA. APPLICATION OF KNOWLEDGE AND UNDERSTANDING CAPABILITIES. TO APPLY THE STUDIED DEFINITIONS, THEOREMS AND RULES FOR SOLVING PROBLEMS. TO USE STRUCTURES AND TOOLS FROM LINEAR ALGEBRA FOR MANAGING MATHEMATICAL PROBLEMS. TO MANAGE 2D AND 3D OBJECTS FROM THE ALGEBRAIC, GEOMETRIC AND ANALYTIC POINT OF VIEW, IN A COORDINATED MANNER TOO. TO APPLY STRUCTURES AND TOOLS FROM PROPOSITIONAL AND PREDICATIVE LOGIC.

THE COURSE PROPOSES THE BASIC ELEMENTS OF LINEAR ALGEBRA, GEOMETRY AND MATHEMATICAL LOGIC. THE EDUCATIONAL OBJECTIVES OF THE COURSE ARE THE ACHIEVEMENT OF THE RESULTS AND PROOF TECHNIQUES, AS WELL AS THE ABILITY TO USE THE RELEVANT COMPUTATIONAL TOOLS.

KNOWLEDGE AND UNDERSTANDING CAPABILITIES.
MATRICES AND LINEAR SYSTEMS. VECTOR AND EUCLIDEAN SPACES. 2D AND 3D ANALYTIC GEOMETRY. PROPOSITIONAL AND PREDICATIVE LOGIC. SOFTWARE TOOLS FOR LINEAR ALGEBRA.

APPLICATION OF KNOWLEDGE AND UNDERSTANDING CAPABILITIES.
TO APPLY THE STUDIED DEFINITIONS, THEOREMS AND RULES FOR SOLVING PROBLEMS. TO USE STRUCTURES AND TOOLS FROM LINEAR ALGEBRA FOR MANAGING MATHEMATICAL PROBLEMS. TO MANAGE 2D AND 3D OBJECTS FROM THE ALGEBRAIC, GEOMETRIC AND ANALYTIC POINT OF VIEW, IN A COORDINATED MANNER TOO. TO APPLY STRUCTURES AND TOOLS FROM PROPOSITIONAL AND PREDICATIVE LOGIC.

	Prerequisites
	FOR A PROFITABLE ACHIEVEMENT OF THE EDUCATIONAL GOALS THE STUDENT IS REQUIRED TO MASTER KNOWLEDGE CONCERNING BASIC MATHEMATICS.

	Contents
	TEACHING UNIT 1: LOGIC OF PROPOSITIONS (HOURS LESSON / EXERCISE / LABORATORY 3/5/0) •1 (1 HOUR LESSON / 1 HOUR EXCERCISE): SINTAX: FUNDAMENTAL OPERATORS./Exercises on conversion of sentences between natural language and logical propositions. Verification of the syntactic correctness of phares. •2 (1 HOUR LESSON / 1 HOUR EXCERCISE): SEMANTICS: TABLES OF TRUTH, VALIDITY AND CONSEQUENCE./Exercises on evaluation of propositions and on verification of correctness of logical consequences. •3 (1 HOUR LESSON / 1 HOUR EXCERCISE): CALCULUS OF NATURAL DEDUCTION: THEOREMS OF DEDUCTION, CORRECTNESS AND COMPLETENESS, FORMAL SYSTEMS./Exercises of logical deduction. •4 (2 HOUR EXCERCISE): Exercises on propositional logic. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: To understand the terms used during the lectures, e.g. logical connective, and, or, xor, if..then, iff, logical consequence, deduction, evaluation function, and of their syntax and priority. APPLYING KNOWLEDGE AND UNDERSTANDING: To recognize correspondence between connectives of the natural language and logical connectives. To recognize the syntactic correctness of a phrase. To construct the syntax tree. To compute the truth value of a proposition. To recognize the correctness of a deduction. To compute a deduction. TEACHING UNIT 2: SETS and Boole Algebras (HOURS LESSON / EXERCISE / LABORATORY 1/3/0) •5 (1 HOUR LESSON / 1 HOUR EXCERCISE): ALGEBRA OF SETS. BOOLE ALGEBRA./Exercises on sets’ representations. •6 (2 HOUR EXCERCISE): Exercises on sets’ operations, both numerical and symbolic. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: To understand the terms used during the lectures, e.g. set, intersection, union, complement, and their properties. APPLYING KNOWLEDGE AND UNDERSTANDING: To be able to describe a set in various ways. To be able to recognize the belonging of an element to a set. To be able to perform a set operation. TEACHING UNIT 3: LOGIC OF PREDICATES (HOURS LESSON / EXERCISE / LABORATORY 2/2/0) •7 (2 HOUR LESSON): PREDICATIVE LANGUAGES: ALPHABET, TERMS, FORMULAS, FREE AND BOUNDED VARIABLES, QUANTIFIERS AND PROOFS. •8 (2 HOUR EXCERCISE): Excises on the management of the quantifiers: transformations of a proposition with a quantifier in an equivalent proposition with the use of the other quantifier, denial of a proposition with quantifiers. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: To understand the terms used during the lectures, e.g. universal quantifier, existential quantifier, and of their properties. APPLYING KNOWLEDGE AND UNDERSTANDING: To be able to interpret and manage a proposition with quantifiers. To be able to prove the falsehood of a proposition by means of counterexamples. TEACHING UNIT 4: ALGEBRIC STRUCTURES (HOURS LESSON / EXERCISE / LABORATORY 1/1/0) -9 (1 HOUR LESSON / 1 HOUR EXCERCISE): GENERAL DEFINITIONS: OPERATIONS AND PROPERTIES. GROUPS. RINGS. FIELDS. / EXAMPLES AND COUNTEREXAMPLES OF THE VARIOUS ALGEBRIC STRUCTURES. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND THE TERMS: OPERATION, GROUP, RING, FIELD AND THEIR PROPERTIES. APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE AN OPERATION, A PROPERTY THAT IS VALID OR NOT FOR AN OPERATION, TO RECOGNIZE THE TYPE OF ALGEBRIC STRUCTURE. TEACHING UNIT 5: MATRICES (HOURS LESSON / EXERCISE / WORKSHOP 2/4/0) -10 (1 HOUR LESSON / 1 HOUR EXCERCISE): DEFINITIONS AND PROPERTIES. REDUCED ROW ECHELON FORM MATRICES. / EXAMPLES AND COUNTEREXAMPLES. EXERCISES OF REDUCTION OF A MATRIX TO ROW ECHELON FORM. -11 (1 HOUR LESSON / 1 HOUR EXERCISE): DETERMINANTS: LAPLACE THEOREM. RANK OF A MATRIX. KRONECKER'S RANK THEOREM. INVERSE OF A MATRIX. / CALCULATION OF DETERMINANTS, RANK AND INVERSE OF A MATRIX. -12 (2 HOURS EXCERCISE): EXERCISES ON MATRICES. USE OF SOFTWARE TOOLS FOR LINEAR ALGEBRA. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND ALL THE TERMS USED DURING LESSON HOURS, E.G. MATRIX (WITH ALL THE PARTICULAR CASES: TRIANGULAR MATRIX, ROW REDUCED MATRIX …), DETERMINANT, RANK, COFACTOR, INVERSE, PIVOT, TRANSPOSED, ADJOINT. TO UNDERSTAND USUAL NOTATIONS AND THEIR MEANING (E.G. MEANING OF SUBSCRIPTS). APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE A TYPE OF MATRIX. TO CALCULATE THE DETERMINANT AND THE RANK OF A MATRIX WITH VARIOUS METHODS. TO USE THE GAUSS ALGORITHM FOR THE REDUCTION OF A MATRIX TO ROW ECHELON FORM. TEACHING UNIT 6: LINEAR SYSTEMS (HOURS LESSON / EXERCISE / WORKSHOP 2/6/0) -13 (1 HOUR LESSON / 1 HOUR EXCERCISE): SYSTEM OF LINEAR EQUATIONS: DEFINITION, ASSOCIATED MATRICES, COMPATIBILITY AND NOT, NUMBER OF SOLUTIONS. / EXERCISES ON SOLUTION OF LINEAR SYSTEMS THROUGH GAUSS’S ELIMINATION METHOD. -14 (1 HOUR LESSON / 1 HOUR EXERCISE): ROUCHÉ-CAPELLI THEOREM. CRAMER THEOREM. / EXERCISES ON SOLUTION OF LINEAR SYSTEMS THROUGH THE CRAMER METHOD, ALSO GENERALIZED IN THE CASE OF RECTANGULAR SYSTEMS. -15 (2 HOURS EXCERCISE): EXERCISES ON NUMERICAL LINEAR SYSTEMS. DISCUSSION OF LINEAR SYSTEMS WITH A PARAMETER. CALCULATION OF A BASIS OF THE SOLUTIONS OF A HOMOGENEOUS LINEAR SYSTEM. USE OF SOFTWARE TOOLS FOR LINEAR ALGEBRA. -16 (2 HOURS EXCERCISE): recap exercises on matrices and linear systems. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND ALL THE TERMS USED DURING LESSON HOURS, E.G. COMPATIBLE, INCOMPATIBLE LINEAR SYSTEM, SOLUTION. TO UNDERSTAND USUAL NOTATIONS (SCALAR AND MATRICIAL) AND THEIR MEANING (E.G. ABILITY TO DISTINGUISH FROM THE CONTEXT BETWEEN THE USE OF THE LETTER 'X' AS A SCALAR VARIABLE OR AS A VECTOR VARIABLE). APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE THE COMPATIBILITY AND NUMBER OF SOLUTIONS OF A LINEAR SYSTEM. TO CALCULATE THE SOLUTIONS OF A LINEAR SYSTEM WITH THE GAUSS METHOD OR WITH THE CRAMER METHOD (GENERALIZED). TO DISCUSS THE COMPATIBILITY AND SOLUTIONS OF A PARAMETRIC SYSTEM. TEACHING UNIT 7: VECTOR SPACES (HOURS LESSON / EXERCISE / LABORATORY 3/7/0) -17 (1 HOUR LESSON / 1 HOUR EXERCISE): THE VECTOR SPACE STRUCTURE. VECTOR SUBSPACES. / EXAMPLES, COUNTEREXAMPLES AND VERIFICATION OF THE CONDITIONS OF VECTOR SPACES AND SUBSPACES. -18 (1 HOUR LESSON / 1 HOUR EXCERCISE): LINEAR DEPENDENCE AND INDEPENDENCE. GENERATORS. GENERATED SUBSPACES. BASES. DIMENSION OF A VECTOR SPACE. LINEAR DEPENDENCE AND INDEPENDENCE CHARACTERIZATIONS FOR NUMERICAL VECTORS. SUFFICIENT CONDITION FOR THE BASES. / EXERCISES ON LINEARLY DEPENDENT AND INDEPENDENT VECTORS AND ON GENERATOR VECTORS. -19 (1 HOUR LESSON / 1 HOUR EXERCISE): STEINITZ LEMMA. BASIS THEOREM. CARTESIAN AND PARAMETRIC REPRESENTATION OF A SUBSPACE. INTERSECTION AND SUM OF SUBSPACES, DIRECT SUM. GRASSMANN THEOREM. / EXERCISES ON CARTESIAN AND PARAMETRIC REPRESENTATIONS OF VECTOR SUBSPACES, BASES AND DIMENSIONS. -20 (2 HOURS EXERCISE): EXERCISES ON NUMERICAL AND PARAMETRIC VECTOR SPACES. USE OF SOFTWARE TOOLS FOR LINEAR ALGEBRA. -21 (2 Hours exercise): recap exercises on vector spaces. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND ALL THE TERMS USED DURING LESSON HOURS, E.G. VECTOR SPACE AND SUBSPACE, BASIS, DIMENSION, CARTESIAN REPRESENTATION, PARAMETRIC REPRESENTATION. TO UNDERSTAND USUAL NOTATIONS (SCALAR AND VECTORIAL) AND THEIR MEANING. APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE IF A SET IS A VECTOR SPACE / SUBSPACE. TO REPRESENT A NUMERIC VECTOR SUBSPACE IN CARTESIAN AND PARAMETRIC FORM. TO CALCULATE THE DIMENSION AND IDENTIFY A BASIS OF A VECTOR SPACE. TO RECOGNIZE WHETHER A SET IS OR NOT A BASIS FOR OF A GIVEN VECTOR SPACE. TO DISCUSS DIMENSION AND BASIS OF A VECTOR SPACE IF THE REPRESENTATION VARIES WITH A PARAMETER. TEACHING UNIT 8: EUCLIDICAL SPACES (HOURS LESSON / EXERCISE / WORKSHOP 2/4/0) -22 (1 HOUR LESSON / 1 HOUR EXERCISE): DEFINITION OF SCALAR PRODUCT. DEFINITION OF REAL EUCLIDEAN VECTOR SPACE. DEFINITION OF NORM. CAUCHY-SCHWARZ INEQUALITY. DEFINITION OF ANGLE. / EXAMPLES AND COUNTEREXAMPLES OF SCALAR PRODUCT. CALCULATION OF NORMS OF VECTORS AND ANGLES BETWEEN VECTORS. -23 (1 HOUR LESSON / 1 HOUR EXERCISE): DEFINITION OF ORTHOGONAL VECTORS AND ORTHOGONAL SUBSPACE. ORTHONORMAL BASES. COMPONENTS IN AN ORTHONORMAL BASIS. ORTHOGONAL PROJECTIONS. GRAM-SCHMIDT THEOREM. / EXERCISES ON ORTHOGONAL SUBSPACES (CALCULATION OF REPRESENTATIONS, DIMENSION, BASIS, BELONGING OF A VECTOR). -24 (2 HOURS EXERCISE): CALCULATION OF ORTHONORMAL BASES THROUGH THE GRAM-SCHMIDT ALGORITHM. CALCULATION OF PROJECTIONS AND ORTHOGONAL COMPONENTS OF A VECTOR ON ANOTHER ONE. CALCULATION OF COMPONENTS OF A VECTOR WITH RESPECT TO AN ORTHONORMAL BASIS. EXERCISES ON ORTHOGONAL SUBSPACES. USE OF SOFTWARE TOOLS FOR LINEAR ALGEBRA. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND ALL THE TERMS USED DURING LESSON HOURS, E.G. EUCLIDEAN SPACE, SCALAR PRODUCT, NORM, ORTHOGONAL SUBSPACE, ORTHONORMAL BASIS. TO UNDERSTAND USUAL NOTATIONS (SCALAR AND VECTORIAL) AND THEIR MEANING. APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE A SCALAR PRODUCT. TO CALCULATE AN ORTHOGONAL SUBSPACE TO A GIVEN ONE (REPRESENTATION, DIMENSION AND BASIS). TO RECOGNIZE IF A VECTOR BELONGS TO THE ORTHOGONAL SUBSPACE TO A GIVEN ONE. TO CALCULATE THE NORM OF A VECTOR. TO CALCULATE THE ANGLE BETWEEN TWO VECTORS. TO RECOGNIZE AND TO CALCULATE AN ORTHONORMAL BASIS AND THE COMPONENTS OF A VECTOR WITH RESPECT TO THAT BASIS. TEACHING UNIT 9: LINEAR APPLICATIONS (HOURS LESSON / EXERCISE / WORKSHOP 2/4/0) -25 (1 HOUR LESSON / 1 HOUR EXCERCISE): DEFINITION OF LINEAR APPLICATION (HOMOMORPHISM), ENDO-, EPI-, MONO-MORPHISM. KERNEL. / EXAMPLES AND COUNTEREXAMPLES OF LINEAR APPLICATIONS. CALCULATION OF THE KERNEL OF A HOMOMORPHISM (CARTESIAN AND PARAMETRIC REPRESENTATION, DIMENSION, BASIS, BELONGING OF VECTORS). -26 (1 HOUR LESSON / 1 HOUR EXERCISE): IMAGE AND ITS GENERATORS. DIMENSION THEOREM. CHARACTERIZATION OF MONOMORPHISMS AND EPIMORPHISMS. MATRIX REPRESENTATION. / CALCULATION OF THE IMAGE OF A HOMOMORPHISM (CARTESIAN AND PARAMETRIC REPRESENTATION, DIMENSION, BASIS, BELONGING OF VECTORS). VERIFICATION OF THE INJECTIVITY AND SURJECTIVITY OF A HOMOMORPHISM. -27 (2 HOURS EXERCISE): VARIOUS EXERCISES ON HOMOMORPHISMS. DISCUSSION OF HOMOMORPHISMS WITH LAWS DEPENDING ON A PARAMETER. USE OF SOFTWARE TOOLS FOR LINEAR ALGEBRA. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND ALL THE TERMS USED DURING LESSON HOURS, E.G. HOMO-, EPI-, MONO-, ISO-MORPHISM, KERNEL, IMAGE. TO UNDERSTAND USUAL NOTATIONS (LAWS AND MATRICES) AND THEIR MEANING. APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE A HOMOMORPHISM. TO MANAGE THE DIFFERENT REPRESENTATIONS OF A HOMOMORPHISM (LAW AND MATRIX). TO CALCULATE KERNEL AND IMAGE (REPRESENTATIONS, DIMENSION AND BASIS, BELONGING OF A VECTOR). TEACHING UNIT 10: DIAGONALIZATION (HOURS LESSON / EXERCISE / WORKSHOP 2/6/0) -28 (1 HOUR LESSON / 1 HOUR EXERCISE): EIGENVALUES AND EIGENVECTORS: DEFINITIONS, CHARACTERISTIC POLYNOMIAL AND EQUATION. EIGENSPACES AND RELATED PROPERTIES. ALGEBRIC AND GEOMETRIC MULTIPLICITY. / EXERCISES ON CALCULATING EIGENVALUES, MULTIPLICITY, EIGENSPACES. -29 (1 HOUR LESSON / 1 HOUR EXCERCISE): SIMPLE AND ORTHOGONAL DIAGONALIZATION: DEFINITIONS FOR MATRICES AND ENDOMORPHISMS. MAIN THEOREM OF DIAGONALIZATION. SPECTRAL THEOREM. / CHECK FOR SIMPLE AND/OR ORTHOGONAL DIAGONALIZABILITY. CALCULATION OF SIMPLE AND ORTHOGONAL DIAGONALIZATION MATRICES. -30 (2 HOURS EXCERCISE): VARIOUS EXERCISES ON DIAGONALIZATION. DISCUSSION OF ORTHOGONAL DIAGONALIZABILITY IN THE PRESENCE OF A PARAMETER. USE OF SOFTWARE TOOLS FOR LINEAR ALGEBRA. -31 (2 HOURS EXCERCISE): recap exercise on homeomorphisms and diagonalization. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND ALL THE TERMS USED DURING LESSON HOURS, E.G. EIGENVALUES AND EIGENVECTORS, CHARACTERISTIC POLYNOMIAL, DIAGONALIZATION. APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE IF A VECTOR IS AN EIGENVECTOR OF A MATRIX. TO CALCULATE EIGENSPACES (REPRESENTATIONS, DIMENSION AND BASIS, BELONGING OF A VECTOR). TO RECOGNIZE IF A MATRIX IS DIAGONALIZABLE AND/OR ORTHOGONALLY DIAGONALIZABLE. TO CALCULATE SIMPLE AND ORTHOGONAL DIAGONALIZATION MATRICES AND THE CORRESPONDING DIAGONAL MATRICES GENERATED. TEACHING UNIT 11: ANALYTICAL GEOMETRY (HOURS LESSON / EXERCISE / WORKSHOP 2/8/0) -32 (1 HOUR LESSON / 1 HOUR EXERCISE): CARTESIAN REFERENCE SYSTEM IN THE PLANE. EQUATION OF THE LINE (ALGEBRIC, PARAMETRIC, SYMMETRIC). PARALLELISM AND ORTHOGONALITY BETWEEN LINES. CONICS: DEFINITION, CLASSIFICATION AND CANONICAL FORM. / EXERCISES ON REPRESENTATIONS OF LINES IN THE PLANE (CONSTRUCTION, BELONGING, CONVERSION BETWEEN DIFFERENT REPRESENTATIONS). -33 (2 HOURS OF EXERCISE): CONSTRUCTION OF LINEAR APPLICATIONS IN THE PLANE (ROTATIONS, REFLECTIONS, EXPANSION AND CONTRACTIONS, DEFORMATIONS). CLASSIFICATION AND REDUCTION TO CANONICAL FORM OF CONICS. USE OF SOFTWARE TOOLS FOR ANALYTICAL GEOMETRY. -34 (1 HOUR LESSON / 1 HOUR EXERCISE): CARTESIAN REFERENCE SYSTEM IN THE SPACE. VECTOR PRODUCT AND MIXED PRODUCT. EQUATION OF THE PLANE (PARAMETRIC AND CARTESIAN). EQUATION OF THE LINE (PARAMETRIC, CARTESIAN, SYMMETRIC). BUNDLES AND STARS OF PLANES. CONDITIONS OF PARALLELISM AND PERPENDICULARITY IN THE SPACE. NON-COPLANAR LINES. / EXERCISES ON REPRESENTATIONS OF LINES AND PLANES IN SPACE (CONSTRUCTION, BELONGING, CONVERSION BETWEEN DIFFERENT REPRESENTATIONS). -35 (2 HOURS EXERCISE): VARIOUS EXERCISES ON THE LINES AND PLANES IN THE SPACE. USE OF SOFTWARE TOOLS FOR ANALYTICAL GEOMETRY. -36 (2 HOURS EXCERCISE): recap exercise on analytic geometry. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND ALL THE TERMS USED DURING LESSON HOURS, E.G. LINES, PLANES, POINTS, EQUATIONS. TO UNDERSTAND USUAL NOTATIONS (SCALAR AND VECTORIAL) AND THEIR MEANING. APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE POINTS, LINES AND CONICS IN THE PLANE. TO RECOGNIZE POINTS, LINES AND PLANES IN SPACE. TO MAKE THE SWITCH BETWEEN DIFFERENT REPRESENTATIONS OF THE SAME GEOMETRIC OBJECT. TO RECOGNIZE THE MUTUAL POSITION OF GEOMETRIC OBJECTS. TOTAL HOURS: 22 HOURS OF LESSON AND 50 HOURS OF EXERCISES

Contents

TEACHING UNIT 1: LOGIC OF PROPOSITIONS
(HOURS LESSON / EXERCISE / LABORATORY 3/5/0)
•1 (1 HOUR LESSON / 1 HOUR EXCERCISE): SINTAX: FUNDAMENTAL OPERATORS./Exercises on conversion of sentences between natural language and logical propositions. Verification of the syntactic correctness of phares.

•2 (1 HOUR LESSON / 1 HOUR EXCERCISE): SEMANTICS: TABLES OF TRUTH, VALIDITY AND CONSEQUENCE./Exercises on evaluation of propositions and on verification of correctness of logical consequences.

•3 (1 HOUR LESSON / 1 HOUR EXCERCISE): CALCULUS OF NATURAL
DEDUCTION: THEOREMS OF DEDUCTION, CORRECTNESS AND COMPLETENESS, FORMAL SYSTEMS./Exercises of logical deduction.

•4 (2 HOUR EXCERCISE): Exercises on propositional logic.

KNOWLEDGE AND UNDERSTANDING CAPABILITIES: To understand the terms used during the lectures, e.g. logical connective, and, or, xor, if..then, iff, logical consequence, deduction, evaluation function, and of their syntax and priority.
APPLYING KNOWLEDGE AND UNDERSTANDING: To recognize correspondence between connectives of the natural language and logical connectives. To recognize the syntactic correctness of a phrase. To construct the syntax tree. To compute the truth value of a proposition. To recognize the correctness of a deduction. To compute a deduction.

TEACHING UNIT 2: SETS and Boole Algebras
(HOURS LESSON / EXERCISE / LABORATORY 1/3/0)
•5 (1 HOUR LESSON / 1 HOUR EXCERCISE): ALGEBRA OF SETS. BOOLE ALGEBRA./Exercises on sets’ representations.

•6 (2 HOUR EXCERCISE): Exercises on sets’ operations, both numerical and symbolic.

KNOWLEDGE AND UNDERSTANDING CAPABILITIES: To understand the terms used during the lectures, e.g. set, intersection, union, complement, and their properties.
APPLYING KNOWLEDGE AND UNDERSTANDING: To be able to describe a set in various ways. To be able to recognize the belonging of an element to a set. To be able to perform a set operation.

TEACHING UNIT 3: LOGIC OF PREDICATES
(HOURS LESSON / EXERCISE / LABORATORY 2/2/0)
•7 (2 HOUR LESSON): PREDICATIVE LANGUAGES: ALPHABET, TERMS, FORMULAS, FREE AND BOUNDED VARIABLES, QUANTIFIERS AND PROOFS.

•8 (2 HOUR EXCERCISE): Excises on the management of the quantifiers: transformations of a proposition with a quantifier in an equivalent proposition with the use of the other quantifier, denial of a proposition with quantifiers.

KNOWLEDGE AND UNDERSTANDING CAPABILITIES: To understand the terms used during the lectures, e.g. universal quantifier, existential quantifier, and of their properties.
APPLYING KNOWLEDGE AND UNDERSTANDING: To be able to interpret and manage a proposition with quantifiers. To be able to prove the falsehood of a proposition by means of counterexamples.

TEACHING UNIT 4: ALGEBRIC STRUCTURES
(HOURS LESSON / EXERCISE / LABORATORY 1/1/0)
-9 (1 HOUR LESSON / 1 HOUR EXCERCISE): GENERAL DEFINITIONS: OPERATIONS AND PROPERTIES. GROUPS. RINGS. FIELDS. / EXAMPLES AND COUNTEREXAMPLES OF THE VARIOUS ALGEBRIC STRUCTURES.
KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND THE TERMS: OPERATION, GROUP, RING, FIELD AND THEIR PROPERTIES.
APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE AN OPERATION, A PROPERTY THAT IS VALID OR NOT FOR AN OPERATION, TO RECOGNIZE THE TYPE OF ALGEBRIC STRUCTURE.
TEACHING UNIT 5: MATRICES
(HOURS LESSON / EXERCISE / WORKSHOP 2/4/0)
-10 (1 HOUR LESSON / 1 HOUR EXCERCISE): DEFINITIONS AND PROPERTIES. REDUCED ROW ECHELON FORM MATRICES. / EXAMPLES AND COUNTEREXAMPLES. EXERCISES OF REDUCTION OF A MATRIX TO ROW ECHELON FORM.

-11 (1 HOUR LESSON / 1 HOUR EXERCISE): DETERMINANTS: LAPLACE THEOREM. RANK OF A MATRIX. KRONECKER'S RANK THEOREM. INVERSE OF A MATRIX. / CALCULATION OF DETERMINANTS, RANK AND INVERSE OF A MATRIX.

-12 (2 HOURS EXCERCISE): EXERCISES ON MATRICES. USE OF SOFTWARE TOOLS FOR LINEAR ALGEBRA.
KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND ALL THE TERMS USED DURING LESSON HOURS, E.G. MATRIX (WITH ALL THE PARTICULAR CASES: TRIANGULAR MATRIX, ROW REDUCED MATRIX …), DETERMINANT, RANK, COFACTOR, INVERSE, PIVOT, TRANSPOSED, ADJOINT. TO UNDERSTAND USUAL NOTATIONS AND THEIR MEANING (E.G. MEANING OF SUBSCRIPTS).
APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE A TYPE OF MATRIX. TO CALCULATE THE DETERMINANT AND THE RANK OF A MATRIX WITH VARIOUS METHODS. TO USE THE GAUSS ALGORITHM FOR THE REDUCTION OF A MATRIX TO ROW ECHELON FORM.
TEACHING UNIT 6: LINEAR SYSTEMS
(HOURS LESSON / EXERCISE / WORKSHOP 2/6/0)
-13 (1 HOUR LESSON / 1 HOUR EXCERCISE): SYSTEM OF LINEAR EQUATIONS: DEFINITION, ASSOCIATED MATRICES, COMPATIBILITY AND NOT, NUMBER OF SOLUTIONS. / EXERCISES ON SOLUTION OF LINEAR SYSTEMS THROUGH GAUSS’S ELIMINATION METHOD.

-14 (1 HOUR LESSON / 1 HOUR EXERCISE): ROUCHÉ-CAPELLI THEOREM. CRAMER THEOREM. / EXERCISES ON SOLUTION OF LINEAR SYSTEMS THROUGH THE CRAMER METHOD, ALSO GENERALIZED IN THE CASE OF RECTANGULAR SYSTEMS.

-15 (2 HOURS EXCERCISE): EXERCISES ON NUMERICAL LINEAR SYSTEMS. DISCUSSION OF LINEAR SYSTEMS WITH A PARAMETER. CALCULATION OF A BASIS OF THE SOLUTIONS OF A HOMOGENEOUS LINEAR SYSTEM. USE OF SOFTWARE TOOLS FOR LINEAR ALGEBRA.

-16 (2 HOURS EXCERCISE): recap exercises on matrices and linear systems.
KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND ALL THE TERMS USED DURING LESSON HOURS, E.G. COMPATIBLE, INCOMPATIBLE LINEAR SYSTEM, SOLUTION. TO UNDERSTAND USUAL NOTATIONS (SCALAR AND MATRICIAL) AND THEIR MEANING (E.G. ABILITY TO DISTINGUISH FROM THE CONTEXT BETWEEN THE USE OF THE LETTER 'X' AS A SCALAR VARIABLE OR AS A VECTOR VARIABLE).
APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE THE COMPATIBILITY AND NUMBER OF SOLUTIONS OF A LINEAR SYSTEM. TO CALCULATE THE SOLUTIONS OF A LINEAR SYSTEM WITH THE GAUSS METHOD OR WITH THE CRAMER METHOD (GENERALIZED). TO DISCUSS THE COMPATIBILITY AND SOLUTIONS OF A PARAMETRIC SYSTEM.
TEACHING UNIT 7: VECTOR SPACES
(HOURS LESSON / EXERCISE / LABORATORY 3/7/0)
-17 (1 HOUR LESSON / 1 HOUR EXERCISE): THE VECTOR SPACE STRUCTURE. VECTOR SUBSPACES. / EXAMPLES, COUNTEREXAMPLES AND VERIFICATION OF THE CONDITIONS OF VECTOR SPACES AND SUBSPACES.

-18 (1 HOUR LESSON / 1 HOUR EXCERCISE): LINEAR DEPENDENCE AND INDEPENDENCE. GENERATORS. GENERATED SUBSPACES. BASES. DIMENSION OF A VECTOR SPACE. LINEAR DEPENDENCE AND INDEPENDENCE CHARACTERIZATIONS FOR NUMERICAL VECTORS. SUFFICIENT CONDITION FOR THE BASES. / EXERCISES ON LINEARLY DEPENDENT AND INDEPENDENT VECTORS AND ON GENERATOR VECTORS.

-19 (1 HOUR LESSON / 1 HOUR EXERCISE): STEINITZ LEMMA. BASIS THEOREM. CARTESIAN AND PARAMETRIC REPRESENTATION OF A SUBSPACE. INTERSECTION AND SUM OF SUBSPACES, DIRECT SUM. GRASSMANN THEOREM. / EXERCISES ON CARTESIAN AND PARAMETRIC REPRESENTATIONS OF VECTOR SUBSPACES, BASES AND DIMENSIONS.

-20 (2 HOURS EXERCISE): EXERCISES ON NUMERICAL AND PARAMETRIC VECTOR SPACES. USE OF SOFTWARE TOOLS FOR LINEAR ALGEBRA.

-21 (2 Hours exercise): recap exercises on vector spaces.
KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND ALL THE TERMS USED DURING LESSON HOURS, E.G. VECTOR SPACE AND SUBSPACE, BASIS, DIMENSION, CARTESIAN REPRESENTATION, PARAMETRIC REPRESENTATION. TO UNDERSTAND USUAL NOTATIONS (SCALAR AND VECTORIAL) AND THEIR MEANING.
APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE IF A SET IS A VECTOR SPACE / SUBSPACE. TO REPRESENT A NUMERIC VECTOR SUBSPACE IN CARTESIAN AND PARAMETRIC FORM. TO CALCULATE THE DIMENSION AND IDENTIFY A BASIS OF A VECTOR SPACE. TO RECOGNIZE WHETHER A SET IS OR NOT A BASIS FOR OF A GIVEN VECTOR SPACE. TO DISCUSS DIMENSION AND BASIS OF A VECTOR SPACE IF THE REPRESENTATION VARIES WITH A PARAMETER.

TEACHING UNIT 8: EUCLIDICAL SPACES
(HOURS LESSON / EXERCISE / WORKSHOP 2/4/0)
-22 (1 HOUR LESSON / 1 HOUR EXERCISE): DEFINITION OF SCALAR PRODUCT. DEFINITION OF REAL EUCLIDEAN VECTOR SPACE. DEFINITION OF NORM. CAUCHY-SCHWARZ INEQUALITY. DEFINITION OF ANGLE. / EXAMPLES AND COUNTEREXAMPLES OF SCALAR PRODUCT. CALCULATION OF NORMS OF VECTORS AND ANGLES BETWEEN VECTORS.

-23 (1 HOUR LESSON / 1 HOUR EXERCISE): DEFINITION OF ORTHOGONAL VECTORS AND ORTHOGONAL SUBSPACE. ORTHONORMAL BASES. COMPONENTS IN AN ORTHONORMAL BASIS. ORTHOGONAL PROJECTIONS. GRAM-SCHMIDT THEOREM. / EXERCISES ON ORTHOGONAL SUBSPACES (CALCULATION OF REPRESENTATIONS, DIMENSION, BASIS, BELONGING OF A VECTOR).

-24 (2 HOURS EXERCISE): CALCULATION OF ORTHONORMAL BASES THROUGH THE GRAM-SCHMIDT ALGORITHM. CALCULATION OF PROJECTIONS AND ORTHOGONAL COMPONENTS OF A VECTOR ON ANOTHER ONE. CALCULATION OF COMPONENTS OF A VECTOR WITH RESPECT TO AN ORTHONORMAL BASIS. EXERCISES ON ORTHOGONAL SUBSPACES. USE OF SOFTWARE TOOLS FOR LINEAR ALGEBRA.
KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND ALL THE TERMS USED DURING LESSON HOURS, E.G. EUCLIDEAN SPACE, SCALAR PRODUCT, NORM, ORTHOGONAL SUBSPACE, ORTHONORMAL BASIS. TO UNDERSTAND USUAL NOTATIONS (SCALAR AND VECTORIAL) AND THEIR MEANING.
APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE A SCALAR PRODUCT. TO CALCULATE AN ORTHOGONAL SUBSPACE TO A GIVEN ONE (REPRESENTATION, DIMENSION AND BASIS). TO RECOGNIZE IF A VECTOR BELONGS TO THE ORTHOGONAL SUBSPACE TO A GIVEN ONE. TO CALCULATE THE NORM OF A VECTOR. TO CALCULATE THE ANGLE BETWEEN TWO VECTORS. TO RECOGNIZE AND TO CALCULATE AN ORTHONORMAL BASIS AND THE COMPONENTS OF A VECTOR WITH RESPECT TO THAT BASIS.
TEACHING UNIT 9: LINEAR APPLICATIONS
(HOURS LESSON / EXERCISE / WORKSHOP 2/4/0)
-25 (1 HOUR LESSON / 1 HOUR EXCERCISE): DEFINITION OF LINEAR APPLICATION (HOMOMORPHISM), ENDO-, EPI-, MONO-MORPHISM. KERNEL. / EXAMPLES AND COUNTEREXAMPLES OF LINEAR APPLICATIONS. CALCULATION OF THE KERNEL OF A HOMOMORPHISM (CARTESIAN AND PARAMETRIC REPRESENTATION, DIMENSION, BASIS, BELONGING OF VECTORS).

-26 (1 HOUR LESSON / 1 HOUR EXERCISE): IMAGE AND ITS GENERATORS. DIMENSION THEOREM. CHARACTERIZATION OF MONOMORPHISMS AND EPIMORPHISMS. MATRIX REPRESENTATION. / CALCULATION OF THE IMAGE OF A HOMOMORPHISM (CARTESIAN AND PARAMETRIC REPRESENTATION, DIMENSION, BASIS, BELONGING OF VECTORS). VERIFICATION OF THE INJECTIVITY AND SURJECTIVITY OF A HOMOMORPHISM.

-27 (2 HOURS EXERCISE): VARIOUS EXERCISES ON HOMOMORPHISMS. DISCUSSION OF HOMOMORPHISMS WITH LAWS DEPENDING ON A PARAMETER. USE OF SOFTWARE TOOLS FOR LINEAR ALGEBRA.
KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND ALL THE TERMS USED DURING LESSON HOURS, E.G. HOMO-, EPI-, MONO-, ISO-MORPHISM, KERNEL, IMAGE. TO UNDERSTAND USUAL NOTATIONS (LAWS AND MATRICES) AND THEIR MEANING.
APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE A HOMOMORPHISM. TO MANAGE THE DIFFERENT REPRESENTATIONS OF A HOMOMORPHISM (LAW AND MATRIX). TO CALCULATE KERNEL AND IMAGE (REPRESENTATIONS, DIMENSION AND BASIS, BELONGING OF A VECTOR).
TEACHING UNIT 10: DIAGONALIZATION
(HOURS LESSON / EXERCISE / WORKSHOP 2/6/0)
-28 (1 HOUR LESSON / 1 HOUR EXERCISE): EIGENVALUES AND EIGENVECTORS: DEFINITIONS, CHARACTERISTIC POLYNOMIAL AND EQUATION. EIGENSPACES AND RELATED PROPERTIES. ALGEBRIC AND GEOMETRIC MULTIPLICITY. / EXERCISES ON CALCULATING EIGENVALUES, MULTIPLICITY, EIGENSPACES.

-29 (1 HOUR LESSON / 1 HOUR EXCERCISE): SIMPLE AND ORTHOGONAL DIAGONALIZATION: DEFINITIONS FOR MATRICES AND ENDOMORPHISMS. MAIN THEOREM OF DIAGONALIZATION. SPECTRAL THEOREM. / CHECK FOR SIMPLE AND/OR ORTHOGONAL DIAGONALIZABILITY. CALCULATION OF SIMPLE AND ORTHOGONAL DIAGONALIZATION MATRICES.

-30 (2 HOURS EXCERCISE): VARIOUS EXERCISES ON DIAGONALIZATION. DISCUSSION OF ORTHOGONAL DIAGONALIZABILITY IN THE PRESENCE OF A PARAMETER. USE OF SOFTWARE TOOLS FOR LINEAR ALGEBRA.

-31 (2 HOURS EXCERCISE): recap exercise on homeomorphisms and diagonalization.
KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND ALL THE TERMS USED DURING LESSON HOURS, E.G. EIGENVALUES AND EIGENVECTORS, CHARACTERISTIC POLYNOMIAL, DIAGONALIZATION.
APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE IF A VECTOR IS AN EIGENVECTOR OF A MATRIX. TO CALCULATE EIGENSPACES (REPRESENTATIONS, DIMENSION AND BASIS, BELONGING OF A VECTOR). TO RECOGNIZE IF A MATRIX IS DIAGONALIZABLE AND/OR ORTHOGONALLY DIAGONALIZABLE. TO CALCULATE SIMPLE AND ORTHOGONAL DIAGONALIZATION MATRICES AND THE CORRESPONDING DIAGONAL MATRICES GENERATED.
TEACHING UNIT 11: ANALYTICAL GEOMETRY
(HOURS LESSON / EXERCISE / WORKSHOP 2/8/0)
-32 (1 HOUR LESSON / 1 HOUR EXERCISE): CARTESIAN REFERENCE SYSTEM IN THE PLANE. EQUATION OF THE LINE (ALGEBRIC, PARAMETRIC, SYMMETRIC). PARALLELISM AND ORTHOGONALITY BETWEEN LINES. CONICS: DEFINITION, CLASSIFICATION AND CANONICAL FORM. / EXERCISES ON REPRESENTATIONS OF LINES IN THE PLANE (CONSTRUCTION, BELONGING, CONVERSION BETWEEN DIFFERENT REPRESENTATIONS).

-33 (2 HOURS OF EXERCISE): CONSTRUCTION OF LINEAR APPLICATIONS IN THE PLANE (ROTATIONS, REFLECTIONS, EXPANSION AND CONTRACTIONS, DEFORMATIONS). CLASSIFICATION AND REDUCTION TO CANONICAL FORM OF CONICS. USE OF SOFTWARE TOOLS FOR ANALYTICAL GEOMETRY.

-34 (1 HOUR LESSON / 1 HOUR EXERCISE): CARTESIAN REFERENCE SYSTEM IN THE SPACE. VECTOR PRODUCT AND MIXED PRODUCT. EQUATION OF THE PLANE (PARAMETRIC AND CARTESIAN). EQUATION OF THE LINE (PARAMETRIC, CARTESIAN, SYMMETRIC). BUNDLES AND STARS OF PLANES. CONDITIONS OF PARALLELISM AND PERPENDICULARITY IN THE SPACE. NON-COPLANAR LINES. / EXERCISES ON REPRESENTATIONS OF LINES AND PLANES IN SPACE (CONSTRUCTION, BELONGING, CONVERSION BETWEEN DIFFERENT REPRESENTATIONS).

-35 (2 HOURS EXERCISE): VARIOUS EXERCISES ON THE LINES AND PLANES IN THE SPACE. USE OF SOFTWARE TOOLS FOR ANALYTICAL GEOMETRY.

-36 (2 HOURS EXCERCISE): recap exercise on analytic geometry.
KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO UNDERSTAND ALL THE TERMS USED DURING LESSON HOURS, E.G. LINES, PLANES, POINTS, EQUATIONS. TO UNDERSTAND USUAL NOTATIONS (SCALAR AND VECTORIAL) AND THEIR MEANING.
APPLYING KNOWLEDGE AND UNDERSTANDING: TO RECOGNIZE POINTS, LINES AND CONICS IN THE PLANE. TO RECOGNIZE POINTS, LINES AND PLANES IN SPACE. TO MAKE THE SWITCH BETWEEN DIFFERENT REPRESENTATIONS OF THE SAME GEOMETRIC OBJECT. TO RECOGNIZE THE MUTUAL POSITION OF GEOMETRIC OBJECTS.
TOTAL HOURS: 22 HOURS OF LESSON AND 50 HOURS OF EXERCISES

	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 EXAM AIMS TO ASSESS: KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS OF THE COURSE; THE MASTERY OF THE MATHEMATICAL LANGUAGE IN THE WRITTEN AND ORAL EXAM; THE CAPABILITY TO PROVE THEOREMS; THE ABILITY TO SOLVE EXERCISES; THE CAPABILITY TO IDENTIFY AND APPLY THE MOST SUITABLE AND EFFICIENT SOLVING METHODS FOR AN EXERCISE; THE CAPABILITY TO APPLY THE ACQUIRED KNOWLEDGE IN SOLVING EXERCISES SHOWN DURING THE LECTURES. THE EXAM TEST AIMS TO ASSESS: KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS SHOWN DURING THE COURSE; MASTERY OF THE MATHEMATICAL LANGUAGE IN WRITTEN AND ORAL TESTS; CAPABILITY OF PROVING THEOREMS; ABILITY OF SOLVING EXERCISES; CAPABILITY OF INDIVIDUATE AND APPLY THE MOST APPROPRIATE AND EFFICIENT METHODS IN SOLVING AN EXERCISE; CAPABILITY TO APPLY THE ACQUIRED KNOWLEDGE IN SOLVING EXERCISES NOT SHOWN DURING THE COURSE. THE EXAM CONSISTS IN A WRITTEN TEST AND AN ORAL INTERVIEW. THE WRITTEN TEST LASTS AT LEAST 120 MINUTES AND IT AIMS TO ASSESS THE CAPABILITIES TO CORRECTLY APPLY THEORETICAL KNOWLEDGE AND TO UNDERSTAND THE PROPOSED PROBLEMS. THE WRITTEN TEST IS MANDATORY TO ACCESS THE ORAL EXAM AND IT CONSISTS IN SOLVING TYPICAL EXERCISES SHOWN DURING THE COURSE AND THE ASSESSMENT WILL TAKE INTO ACCOUNT THE SOLVING PROBLEM APPROACH TO THE PROPOSED PROBLEMS AND THE CLARITY AND THE COMPLETENESS OF THE PRESENTATION. IN CASE OF SUCCESS IN THE WRITTEN TEST, A GRADE WILL BE ASSIGNED IN TERMS OF QUALITATIVE SLOTS. THE TEST IS TAKEN BEFORE THE ORAL EXAM AND IT SUCCESSES IF THE SCORE IS GREATER OR EQUAL TO THE ESTABLISHED MINIMUM LEVEL. THE ORAL INTERVIEW MAINLY AIMS TO ASSESS THE LEVEL OF KNOWLEDGE OF ALL THE TOPICS OF THE COURSE, AND IT FOCUSES ON DEFINITIONS, STATEMENTS AND PROOFS OF THEOREMS, UNDERSTANDING OF EXERCISES SOLVING PROCEDURES. THE FINAL GRADE, EXPRESSED AS PART OF 30 WITH POSSIBLE LAUDE, IS DETERMINED STARTING FROM THE GRADE OF THE WRITTEN TEST (THAT HAS A PREVALENT WEIGHT), REGULATING IT IN EXCESS OR DEFECT ACCORDING TO THE ORAL INTERVIEW. BESIDES THE REFERENCE TEXTS, FURTHER TEACHING MATERIAL TOGETHER WITH EXAMPLES OF WRITTEN TESTS CAN BE FOUND IN THE UNIVERSITY ELEARNING PLATFORM AREA DEVOTED TO THE COURSE.

Verification of learning

THE EXAM AIMS TO ASSESS: KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS OF THE COURSE; THE MASTERY OF THE MATHEMATICAL LANGUAGE IN THE WRITTEN AND ORAL EXAM; THE CAPABILITY TO PROVE THEOREMS; THE ABILITY TO SOLVE EXERCISES; THE CAPABILITY TO IDENTIFY AND APPLY THE MOST SUITABLE AND EFFICIENT SOLVING METHODS FOR AN EXERCISE; THE CAPABILITY TO APPLY THE ACQUIRED KNOWLEDGE IN SOLVING EXERCISES SHOWN DURING THE LECTURES.
THE EXAM TEST AIMS TO ASSESS: KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS SHOWN DURING THE COURSE; MASTERY OF THE MATHEMATICAL LANGUAGE IN WRITTEN AND ORAL TESTS; CAPABILITY OF PROVING THEOREMS; ABILITY OF SOLVING EXERCISES; CAPABILITY OF INDIVIDUATE AND APPLY THE MOST APPROPRIATE AND EFFICIENT METHODS IN SOLVING AN EXERCISE; CAPABILITY TO APPLY THE ACQUIRED KNOWLEDGE IN SOLVING EXERCISES NOT SHOWN DURING THE COURSE.
THE EXAM CONSISTS IN A WRITTEN TEST AND AN ORAL INTERVIEW.
THE WRITTEN TEST LASTS AT LEAST 120 MINUTES AND IT AIMS TO ASSESS THE CAPABILITIES TO CORRECTLY APPLY THEORETICAL KNOWLEDGE AND TO UNDERSTAND THE PROPOSED PROBLEMS. THE WRITTEN TEST IS MANDATORY TO ACCESS THE ORAL EXAM AND IT CONSISTS IN SOLVING TYPICAL EXERCISES SHOWN DURING THE COURSE AND THE ASSESSMENT WILL TAKE INTO ACCOUNT THE SOLVING PROBLEM APPROACH TO THE PROPOSED PROBLEMS AND THE CLARITY AND THE COMPLETENESS OF THE PRESENTATION. IN CASE OF SUCCESS IN THE WRITTEN TEST, A GRADE WILL BE ASSIGNED IN TERMS OF QUALITATIVE SLOTS. THE TEST IS TAKEN BEFORE THE ORAL EXAM AND IT SUCCESSES IF THE SCORE IS GREATER OR EQUAL TO THE ESTABLISHED MINIMUM LEVEL.
THE ORAL INTERVIEW MAINLY AIMS TO ASSESS THE LEVEL OF KNOWLEDGE OF ALL THE TOPICS OF THE COURSE, AND IT FOCUSES ON DEFINITIONS, STATEMENTS AND PROOFS OF THEOREMS, UNDERSTANDING OF EXERCISES SOLVING PROCEDURES.
THE FINAL GRADE, EXPRESSED AS PART OF 30 WITH POSSIBLE LAUDE, IS DETERMINED STARTING FROM THE GRADE OF THE WRITTEN TEST (THAT HAS A PREVALENT WEIGHT), REGULATING IT IN EXCESS OR DEFECT ACCORDING TO THE ORAL INTERVIEW.
BESIDES THE REFERENCE TEXTS, FURTHER TEACHING MATERIAL TOGETHER WITH EXAMPLES OF WRITTEN TESTS CAN BE FOUND IN THE UNIVERSITY ELEARNING PLATFORM AREA DEVOTED TO THE COURSE.

	Texts
	G. ALBANO, LA PROVA SCRITTA DI GEOMETRIA: TRA TEORIA E PRATICA, MAGGIOLI (2013). G. ALBANO, C. D'APICE, S. SALERNO, ALGEBRA LINEARE, CUES (2002). G. LOLLI, LOGICA MATEMATICA, DISPENSE ON LINE HTTP://HOMEPAGE.SNS.IT/LOLLI/DISPENSE07.HTM. D. PALLADINO. CORSO DI LOGICA. INTRODUZIONE AL CALCOLO DEI PREDICATI. ED. CAROCCI (2021). SUPPLEMENTARY TEACHING MATERIAL WILL BE AVAILABLE ON THE UNIVERSITY E-LEARNING PLATFORM (HTTP://ELEARNING.UNISA.IT) ACCESSIBLE TO STUDENTS USING THEIR OWN UNIVERSITY CREDENTIALS.

Texts

G. ALBANO, LA PROVA SCRITTA DI GEOMETRIA: TRA TEORIA E PRATICA, MAGGIOLI (2013).
G. ALBANO, C. D'APICE, S. SALERNO, ALGEBRA LINEARE, CUES (2002).
G. LOLLI, LOGICA MATEMATICA, DISPENSE ON LINE HTTP://HOMEPAGE.SNS.IT/LOLLI/DISPENSE07.HTM.
D. PALLADINO. CORSO DI LOGICA. INTRODUZIONE AL CALCOLO DEI PREDICATI. ED. CAROCCI (2021).

SUPPLEMENTARY TEACHING MATERIAL WILL BE AVAILABLE ON THE UNIVERSITY E-LEARNING PLATFORM (HTTP://ELEARNING.UNISA.IT) ACCESSIBLE TO STUDENTS USING THEIR OWN UNIVERSITY CREDENTIALS.

	More Information
	THE COURSE IS HELD IN ITALIAN

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