Curriculum
Canali
Programme
Matrices and operations between matrices. Linear systems and their resolution.
Prerequisiti
noneCore Documentation
Giulia Maria Piacentini CattaneoMatematica discreta e applicazioni
Zanichelli 2008
Reference Bibliography
Nicholson Algebra lineare McGraw-Hill 2001Attendance
recommended attendanceType of evaluation
written testProgramme
1. Linear equations and numbersSystems 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, exercisesAttendance
Not mandatoryType of evaluation
A written test with questions regarding the theory and exercises, plus a possible oral test.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
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
noneCore Documentation
Giulia Maria Piacentini CattaneoMatematica discreta e applicazioni
Zanichelli 2008
Attendance
recommended attendanceType of evaluation
written testMutuazione: 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 numbersSystems 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, exercisesAttendance
Not mandatoryType 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
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
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
noneCore Documentation
Giulia Maria Piacentini CattaneoMatematica discreta e applicazioni
Zanichelli 2008
Attendance
recommended attendanceType of evaluation
written testMutuazione: 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 numbersSystems 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, exercisesAttendance
Not mandatoryType 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
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
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
noneCore Documentation
Giulia Maria Piacentini CattaneoMatematica discreta e applicazioni
Zanichelli 2008
Attendance
recommended attendanceType of evaluation
written testMutuazione: 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 numbersSystems 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, exercisesAttendance
Not mandatoryType 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
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.