| GEOMETRY AND COMBINATORICS
(objectives)
The course aims to provide an introduction to those aspects of linear and discrete mathematics needed in science and engineering.
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Code
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20810098 |
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Language
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ITA |
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Type of certificate
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Profit certificate
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| Module: GEOMETRY AND COMBINATORICS
(objectives)
The course aims to provide an introduction to those aspects of linear and discrete mathematics needed in science and engineering.
|
|
Code
|
20810098-1 |
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Language
|
ITA |
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Type of certificate
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Profit certificate
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Credits
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6
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Scientific Disciplinary Sector Code
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MAT/03
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Contact Hours
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54
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Type of Activity
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Basic compulsory activities
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Group: CANALE 1
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Derived from
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20810098-1 GEOMETRY AND COMBINATORICS in Computer science and engineering L-8 CANALE 1 MEROLA FRANCESCA
(syllabus)
Elements of set theory. Maps between sets: invective, surjective, bijective maps.
Elements of propositional logic, truth tables. Equivalence and order relations.
Combinatorics. Binomial coefficients and binomial theorem. Permutations. The Integers: divisibility, GCD and Euclidean algorithm, Bézout identity, linear congruences.
Baiscs of algebraic structures: permutation groups, abstract groups, polynomials and finite fields.
Elements of graph theory.Lattices and Boolean algebras
(reference books)
Giulia Maria Piacentini Cattaneo
Matematica discreta e applicazioni
Zanichelli 2008
|
|
Dates of beginning and end of teaching activities
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From 27/09/2021 to 21/01/2022 |
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Delivery mode
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Traditional
At a distance
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Attendance
|
not mandatory
|
|
Evaluation methods
|
Written test
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Group: CANALE 2
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Derived from
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20810098-1 GEOMETRY AND COMBINATORICS in Computer science and engineering L-8 CANALE 2 SALVITTI DARIO
(syllabus)
Elements of set theory. Maps between sets: invective, surjective, bijective maps.
Elements of propositional logic, truth tables. Equivalence and order relations.
Combinatorics. Binomial coefficients and binomial theorem. Permutations. The Integers: divisibility, GCD and Euclidean algorithm, Bézout identity, linear congruences.
Baiscs of algebraic structures: permutation groups, abstract groups, polynomials and finite fields.
Elements of graph theory.Lattices and Boolean algebras
(reference books)
Giulia Maria Piacentini Cattaneo
Matematica discreta e applicazioni
Zanichelli 2008
|
|
Dates of beginning and end of teaching activities
|
From 27/09/2021 to 21/01/2022 |
|
Delivery mode
|
Traditional
At a distance
|
|
Attendance
|
not mandatory
|
|
Evaluation methods
|
Written test
|
|
|
| Module: GEOMETRY AND COMBINATORICS
(objectives)
The course aims to provide an introduction to those aspects of linear and discrete mathematics needed in science and engineering.
|
|
Code
|
20810098-2 |
|
Language
|
ITA |
|
Type of certificate
|
Profit certificate
|
|
Credits
|
6
|
|
Scientific Disciplinary Sector Code
|
MAT/09
|
|
Contact Hours
|
54
|
|
Type of Activity
|
Basic compulsory activities
|
Group: CANALE I
|
Derived from
|
20810098-2 GEOMETRY AND COMBINATORICS in Computer science and engineering L-8 CANALE 1 D'ARIANO ANDREA
(syllabus)
1. Linear equations and numbers
Systems of linear equations. Represent linear systems with matrix equations. Equivalent systems. Natural, integers, rational, real numbers and their properties. Pills of Graph Theory: subsets and set subtraction.
2. Matrixes and sets
Matrix with real coefficients. Square, triangular, diagonal matrixes. Transposed and symmetric matrix. Pills of Graph Theory: union and intersection.
3. Vector spaces and matrices
Matrix addition and scalar product and their properties.
4. Matrix product
Matrix product and its mathematical properties: associative and distributive. Example of how matrix product does not satisfy the commutative and simplification properties. Matrices and linear systems
5. Determinants
Formal definition of determinant and its properties. Transposed and triangular matrix determinants. Binet’s theorem.
6. Inverse of a matrix
Identity matrix. Inverse of a matrix and its properties. Cramer’s theorem.
7. Matrix rank
Definition and properties. Minor of a matrix.
8. System of linear equations.
Definition. Rouché-Capelli’s theorem. Rouché-Capelli’s method to solve a linear system.
9. Gaussian elimination and how to apply the Gaussian elimination
Elementary operation. Determinant and rank computations.
10. Geometric vectors
Vectors in 2 dimensions. Sum and scalar product of vectors. Vectors in 3 dimensions. Linear combination. Linear dependence and independence of vectors.
11. Vector spaces and Linear subspaces
Definition, examples and properties.
12. How to generate linear subspaces
Linear combination and generators.
13. Linear dependence and independence
14. Basis of a linear subspace
Basis. Dimensions. Solution set of an homogeneous system and its dimensions.
15. Vector subspaces sum and intersection
Grassmann formula.
16. Affine space
Lines in planes and space. Affine subspace.
17. Homomorphisms
Homomorphism between vector spaces. Represent an homomorphism as a matrix.
18. Image of a matrix
Properties of an homomorphism image and how to compute it. Surjective homomorphism.
19. Kernel of a matrix
Properties of an homomorphism kernel and how to compute it. Injectivity homomorphism.
20. Endomorphisms
Matrix representation of the endomorphisms. Change of basis.
21. Eigenvalues and Eigenvectors
Definition and properties. Eigenspaces. Characteristic polynomial. Diagonalizable matrix.
22. Diagonalizing a matrix
Rules and examples.
(reference books)
G. Accascina e V. Monti,
"Geometria"
|
|
Dates of beginning and end of teaching activities
|
From 27/09/2021 to 21/01/2022 |
|
Delivery mode
|
Traditional
|
|
Attendance
|
not mandatory
|
|
Evaluation methods
|
Written test
Oral exam
|
Group: CANALE 2
|
Derived from
|
20810098-2 GEOMETRY AND COMBINATORICS in Computer science and engineering L-8 CANALE 2 SAMA' MARCELLA
(syllabus)
1. Linear equations and numbers
Systems of linear equations. Represent linear systems with matrix equations. Equivalent systems. Natural, integers, rational, real numbers and their properties. Pills of Graph Theory: subsets and set subtraction.
2. Matrixes and sets
Matrix with real coefficients. Square, triangular, diagonal matrixes. Transposed and symmetric matrix. Pills of Graph Theory: union and intersection.
3. Vector spaces and matrices
Matrix addition and scalar product and their properties.
4. Matrix product
Matrix product and its mathematical properties: associative and distributive. Example of how matrix product does not satisfy the commutative and simplification properties. Matrices and linear systems
5. Determinants
Formal definition of determinant and its properties. Transposed and triangular matrix determinants. Binet’s theorem.
6. Inverse of a matrix
Identity matrix. Inverse of a matrix and its properties. Cramer’s theorem.
7. Matrix rank
Definition and properties. Minor of a matrix.
8. System of linear equations.
Definition. Rouché-Capelli’s theorem. Rouché-Capelli’s method to solve a linear system.
9. Gaussian elimination and how to apply the Gaussian elimination
Elementary operation. Determinant and rank computations.
10. Geometric vectors
Vectors in 2 dimensions. Sum and scalar product of vectors. Vectors in 3 dimensions. Linear combination. Linear dependence and independence of vectors.
11. Vector spaces and Linear subspaces
Definition, examples and properties.
12. How to generate linear subspaces
Linear combination and generators.
13. Linear dependence and independence
14. Basis of a linear subspace
Basis. Dimensions. Solution set of an homogeneous system and its dimensions.
15. Vector subspaces sum and intersection
Grassmann formula.
16. Affine space
Lines in planes and space. Affine subspace.
17. Homomorphisms
Homomorphism between vector spaces. Represent an homomorphism as a matrix.
18. Image of a matrix
Properties of an homomorphism image and how to compute it. Surjective homomorphism.
19. Kernel of a matrix
Properties of an homomorphism kernel and how to compute it. Injectivity homomorphism.
20. Endomorphisms
Matrix representation of the endomorphisms. Change of basis.
21. Eigenvalues and Eigenvectors
Definition and properties. Eigenspaces. Characteristic polynomial. Diagonalizable matrix.
22. Diagonalizing a matrix
Rules and examples.
(reference books)
G. Accascina e V. Monti,
"Geometria"
|
|
Dates of beginning and end of teaching activities
|
From 27/09/2021 to 21/01/2022 |
|
Delivery mode
|
Traditional
|
|
Attendance
|
not mandatory
|
|
Evaluation methods
|
Written test
Oral exam
|
|
|
|
Università Roma Tre