20810518-2 - GEOMETRY AND COMBINATORICS

The course aims to provide an introduction to those aspects of linear algebra and geometry needed in science and engineering.

Curriculum

Canali

teacher profile | teaching materials

Programme


Matrices and operations between matrices. Linear systems and their resolution.

Prerequisiti

none

Core Documentation

Giulia Maria Piacentini Cattaneo
Matematica discreta e applicazioni
Zanichelli 2008


Reference Bibliography

Nicholson Algebra lineare McGraw-Hill 2001

Attendance

recommended attendance

Type of evaluation

written test

teacher profile | teaching materials

Programme

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.



Core Documentation

G. Accascina e V. Monti,
"Geometria"


Type of delivery of the course

frontal lessons, exercises

Attendance

Not mandatory

Type of evaluation

A written test with questions regarding the theory and exercises, plus a possible oral test.

teacher profile | teaching materials

Type of evaluation

Grading criteria: Student assessment is based on the Dublin Descriptors and takes into account the level of knowledge, understanding, and critical processing of the topics covered in the course. Students who demonstrate that they have acquired the fundamental concepts and are able to apply them in simple contexts will receive a passing grade. Average grades will be awarded to those who, in addition to mastering the basics, show the ability to analyze and independently elaborate on more complex concepts. The highest grades are reserved for students who demonstrate in-depth understanding, synthesis skills, critical thinking, and autonomy in solving problems, including new ones or those not explicitly addressed during the course.

Canali

teacher profile | teaching materials

Mutuazione: 20810518-2 GEOMETRIA E COMBINATORIA II MODULO in Ingegneria Informatica e dell'Intelligenza Artificiale L-8 R CANALE 1 MEROLA FRANCESCA, SAMA' MARCELLA, TESSITORE MARTA LEONINA

Programme


Matrices and operations between matrices. Linear systems and their resolution.

Prerequisiti

none

Core Documentation

Giulia Maria Piacentini Cattaneo
Matematica discreta e applicazioni
Zanichelli 2008


Attendance

recommended attendance

Type of evaluation

written test

teacher profile | teaching materials

Mutuazione: 20810518-2 GEOMETRIA E COMBINATORIA II MODULO in Ingegneria Informatica e dell'Intelligenza Artificiale L-8 R CANALE 1 MEROLA FRANCESCA, SAMA' MARCELLA, TESSITORE MARTA LEONINA

Programme

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.



Core Documentation

G. Accascina e V. Monti,
"Geometria"


Type of delivery of the course

frontal lessons, exercises

Attendance

Not mandatory

Type of evaluation

A written test with questions regarding the theory and exercises, plus a possible oral test.

Mutuazione: 20810518-2 GEOMETRIA E COMBINATORIA II MODULO in Ingegneria Informatica e dell'Intelligenza Artificiale L-8 R CANALE 1 MEROLA FRANCESCA, SAMA' MARCELLA, TESSITORE MARTA LEONINA

teacher profile | teaching materials

Mutuazione: 20810518-2 GEOMETRIA E COMBINATORIA II MODULO in Ingegneria Informatica e dell'Intelligenza Artificiale L-8 R CANALE 2 SALVITTI DARIO

Type of evaluation

Grading criteria: Student assessment is based on the Dublin Descriptors and takes into account the level of knowledge, understanding, and critical processing of the topics covered in the course. Students who demonstrate that they have acquired the fundamental concepts and are able to apply them in simple contexts will receive a passing grade. Average grades will be awarded to those who, in addition to mastering the basics, show the ability to analyze and independently elaborate on more complex concepts. The highest grades are reserved for students who demonstrate in-depth understanding, synthesis skills, critical thinking, and autonomy in solving problems, including new ones or those not explicitly addressed during the course.

Canali

teacher profile | teaching materials

Mutuazione: 20810518-2 GEOMETRIA E COMBINATORIA II MODULO in Ingegneria Informatica e dell'Intelligenza Artificiale L-8 R CANALE 1 MEROLA FRANCESCA, SAMA' MARCELLA, TESSITORE MARTA LEONINA

Programme


Matrices and operations between matrices. Linear systems and their resolution.

Prerequisiti

none

Core Documentation

Giulia Maria Piacentini Cattaneo
Matematica discreta e applicazioni
Zanichelli 2008


Attendance

recommended attendance

Type of evaluation

written test

teacher profile | teaching materials

Mutuazione: 20810518-2 GEOMETRIA E COMBINATORIA II MODULO in Ingegneria Informatica e dell'Intelligenza Artificiale L-8 R CANALE 1 MEROLA FRANCESCA, SAMA' MARCELLA, TESSITORE MARTA LEONINA

Programme

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.



Core Documentation

G. Accascina e V. Monti,
"Geometria"


Type of delivery of the course

frontal lessons, exercises

Attendance

Not mandatory

Type of evaluation

A written test with questions regarding the theory and exercises, plus a possible oral test.

Mutuazione: 20810518-2 GEOMETRIA E COMBINATORIA II MODULO in Ingegneria Informatica e dell'Intelligenza Artificiale L-8 R CANALE 1 MEROLA FRANCESCA, SAMA' MARCELLA, TESSITORE MARTA LEONINA

teacher profile | teaching materials

Mutuazione: 20810518-2 GEOMETRIA E COMBINATORIA II MODULO in Ingegneria Informatica e dell'Intelligenza Artificiale L-8 R CANALE 2 SALVITTI DARIO

Type of evaluation

Grading criteria: Student assessment is based on the Dublin Descriptors and takes into account the level of knowledge, understanding, and critical processing of the topics covered in the course. Students who demonstrate that they have acquired the fundamental concepts and are able to apply them in simple contexts will receive a passing grade. Average grades will be awarded to those who, in addition to mastering the basics, show the ability to analyze and independently elaborate on more complex concepts. The highest grades are reserved for students who demonstrate in-depth understanding, synthesis skills, critical thinking, and autonomy in solving problems, including new ones or those not explicitly addressed during the course.

Canali

teacher profile | teaching materials

Mutuazione: 20810518-2 GEOMETRIA E COMBINATORIA II MODULO in Ingegneria Informatica e dell'Intelligenza Artificiale L-8 R CANALE 1 MEROLA FRANCESCA, SAMA' MARCELLA, TESSITORE MARTA LEONINA

Programme


Matrices and operations between matrices. Linear systems and their resolution.

Prerequisiti

none

Core Documentation

Giulia Maria Piacentini Cattaneo
Matematica discreta e applicazioni
Zanichelli 2008


Attendance

recommended attendance

Type of evaluation

written test

teacher profile | teaching materials

Mutuazione: 20810518-2 GEOMETRIA E COMBINATORIA II MODULO in Ingegneria Informatica e dell'Intelligenza Artificiale L-8 R CANALE 1 MEROLA FRANCESCA, SAMA' MARCELLA, TESSITORE MARTA LEONINA

Programme

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.



Core Documentation

G. Accascina e V. Monti,
"Geometria"


Type of delivery of the course

frontal lessons, exercises

Attendance

Not mandatory

Type of evaluation

A written test with questions regarding the theory and exercises, plus a possible oral test.

Mutuazione: 20810518-2 GEOMETRIA E COMBINATORIA II MODULO in Ingegneria Informatica e dell'Intelligenza Artificiale L-8 R CANALE 1 MEROLA FRANCESCA, SAMA' MARCELLA, TESSITORE MARTA LEONINA

teacher profile | teaching materials

Mutuazione: 20810518-2 GEOMETRIA E COMBINATORIA II MODULO in Ingegneria Informatica e dell'Intelligenza Artificiale L-8 R CANALE 2 SALVITTI DARIO

Type of evaluation

Grading criteria: Student assessment is based on the Dublin Descriptors and takes into account the level of knowledge, understanding, and critical processing of the topics covered in the course. Students who demonstrate that they have acquired the fundamental concepts and are able to apply them in simple contexts will receive a passing grade. Average grades will be awarded to those who, in addition to mastering the basics, show the ability to analyze and independently elaborate on more complex concepts. The highest grades are reserved for students who demonstrate in-depth understanding, synthesis skills, critical thinking, and autonomy in solving problems, including new ones or those not explicitly addressed during the course.