Provide knowledge of mathematics to understand its application to management. Especially some elements of linear algebra, geometry and graph theory are crucial to the study of management science and engineering.
The various topics will be addressed with a practical approach, moving from the treatment of specific problems to the general case and encouraging the active participation of students to make them acquire the concepts more easily. At the end of the course the student will be able to use the tools of linear algebra and graph theory to address the typical mathematical problems of the subsequent courses.
Systems of linear equations, vectors, matrices, linear and affine spaces, determinants, rank of a matrix, inverse matrix, homomorphisms and diagonalizations, eigenvalues and eigenvectors. Graph theory, directed and undirected graphs, shortest paths, spanning trees, flow networks.
The various topics will be addressed with a practical approach, moving from the treatment of specific problems to the general case and encouraging the active participation of students to make them acquire the concepts more easily. At the end of the course the student will be able to use the tools of linear algebra and graph theory to address the typical mathematical problems of the subsequent courses.
Systems of linear equations, vectors, matrices, linear and affine spaces, determinants, rank of a matrix, inverse matrix, homomorphisms and diagonalizations, eigenvalues and eigenvectors. Graph theory, directed and undirected graphs, shortest paths, spanning trees, flow networks.
Curriculum
teacher profile teaching materials
Linear equations systems. Matrix associated with a linear system. Equivalent systems. Natural, integer, rational numbers, real numbers and their property. Recall of set theory: inclusion of sets, difference between sets.
2. Matrices and sets
Matrices with real coefficients. Square, triangular, diagonal matrices. Transpose of a matrix and symmetric matrices. Recall of set theory: union and intersection of sets.
3. The vector space of the matrices
Addition between matrices and its properties. Multiplying a scalar for a matrix and its properties.
4. Product between matrices
Product between matrices with compatible dimensions. Properties of the product: associative property and distributive property. Examples showing that product between matrices does not satisfy the commutative property and the simplification property. Matrices and linear systems.
5. Determinants
Definition by induction of the determinant when using the first-row development. Determinant property: development according to any row or column, determinant of the transposed matrix, determinant of a triangular matrix. Binet theorem.
6. Reverse matrix
Unit matrix. Reverse matrix. Inverse property. Cramer's theorem.
7. Rank of a matrix
Definition. Property of the rank. Minors of a matrix. Theorem of Kronecker.
8. Linear equation systems
Definitions. Rouché-Capelli theorem. Rouché-Capelli method for solving a linear system.
9. Gauss method
10. Applications of Gauss method
Basic operations. Calculation of the determinant. Calculation of the rank.
11. Geometric vectors
Plan vectors. Addition between vectors. Product of a vector for a scalar. Space vectors. Lines and planes for the origin. Average point.
12. Linear combinations of geometric vectors
Linear combinations. Linearly dependent and independent vectors. Characterization of linearly independent vectors in V2(O) and V3(O).
13. Vector spaces on the real numbers
Definition of vector spaces. Examples of vector spaces. Basic properties of vector spaces.
14. Vector subspaces
Definition of vector spaces. Subspaces of V2(O) and V3(O).
15. Generators of vector spaces
Linear combinations and generators.
16. Linear dependency and independency
17. Basis of vector spaces
Basis. Dimension. Dimension of the set of solutions of a homogeneous system. Dimension of subspaces. Calculation of dimensions and basis.
18. Intersection and sum of subspaces
Intersection of vector subspaces. Sum of vector subspaces. Grassmann's formula.
19. Affine subspaces
The lines of the plane and of the space. Space planes. Affine subspaces. Set of system solutions.
20. Homomorphisms
Homomorphisms between vector spaces. Matrix associated with a homomorphism. Homomorphism associated with a matrix.
21. Image
Property of the image of a homomorphism. Calculation of the image of a homomorphism. Condition of surjectivity of a homomorphism.
22. Kernel
Property of the kernel of a homomorphism. Calculation of the kernel of a homomorphism. Injectivity condition of a homomorphism.
23. Endomorphisms
Matrix associated with an endomorphism. Change of basis.
24. Eigenvalues and eigenvectors
Definitions and basic properties. Eigenspaces. Characteristic polynomial. Diagonalizable matrices.
25. Diagonalization
Diagonalizability conditions. Diagonalization procedure.
26. Graph optimization:
Maximum flow. Shortest path. Minimum spanning tree.
* This book is available for free at the following link: http://www.dmmm.uniroma1.it/accascinamonti/geogest/Geometria.pdf
Caramia, Giordani, Guerriero, Musmanno, Pacciarelli, "Ricerca Operativa", Isedi, Italia, 2014.
Programme
1. Linear equations and numbersLinear equations systems. Matrix associated with a linear system. Equivalent systems. Natural, integer, rational numbers, real numbers and their property. Recall of set theory: inclusion of sets, difference between sets.
2. Matrices and sets
Matrices with real coefficients. Square, triangular, diagonal matrices. Transpose of a matrix and symmetric matrices. Recall of set theory: union and intersection of sets.
3. The vector space of the matrices
Addition between matrices and its properties. Multiplying a scalar for a matrix and its properties.
4. Product between matrices
Product between matrices with compatible dimensions. Properties of the product: associative property and distributive property. Examples showing that product between matrices does not satisfy the commutative property and the simplification property. Matrices and linear systems.
5. Determinants
Definition by induction of the determinant when using the first-row development. Determinant property: development according to any row or column, determinant of the transposed matrix, determinant of a triangular matrix. Binet theorem.
6. Reverse matrix
Unit matrix. Reverse matrix. Inverse property. Cramer's theorem.
7. Rank of a matrix
Definition. Property of the rank. Minors of a matrix. Theorem of Kronecker.
8. Linear equation systems
Definitions. Rouché-Capelli theorem. Rouché-Capelli method for solving a linear system.
9. Gauss method
10. Applications of Gauss method
Basic operations. Calculation of the determinant. Calculation of the rank.
11. Geometric vectors
Plan vectors. Addition between vectors. Product of a vector for a scalar. Space vectors. Lines and planes for the origin. Average point.
12. Linear combinations of geometric vectors
Linear combinations. Linearly dependent and independent vectors. Characterization of linearly independent vectors in V2(O) and V3(O).
13. Vector spaces on the real numbers
Definition of vector spaces. Examples of vector spaces. Basic properties of vector spaces.
14. Vector subspaces
Definition of vector spaces. Subspaces of V2(O) and V3(O).
15. Generators of vector spaces
Linear combinations and generators.
16. Linear dependency and independency
17. Basis of vector spaces
Basis. Dimension. Dimension of the set of solutions of a homogeneous system. Dimension of subspaces. Calculation of dimensions and basis.
18. Intersection and sum of subspaces
Intersection of vector subspaces. Sum of vector subspaces. Grassmann's formula.
19. Affine subspaces
The lines of the plane and of the space. Space planes. Affine subspaces. Set of system solutions.
20. Homomorphisms
Homomorphisms between vector spaces. Matrix associated with a homomorphism. Homomorphism associated with a matrix.
21. Image
Property of the image of a homomorphism. Calculation of the image of a homomorphism. Condition of surjectivity of a homomorphism.
22. Kernel
Property of the kernel of a homomorphism. Calculation of the kernel of a homomorphism. Injectivity condition of a homomorphism.
23. Endomorphisms
Matrix associated with an endomorphism. Change of basis.
24. Eigenvalues and eigenvectors
Definitions and basic properties. Eigenspaces. Characteristic polynomial. Diagonalizable matrices.
25. Diagonalization
Diagonalizability conditions. Diagonalization procedure.
26. Graph optimization:
Maximum flow. Shortest path. Minimum spanning tree.
Core Documentation
G. Accascina and V. Monti, Geometry** This book is available for free at the following link: http://www.dmmm.uniroma1.it/accascinamonti/geogest/Geometria.pdf
Caramia, Giordani, Guerriero, Musmanno, Pacciarelli, "Ricerca Operativa", Isedi, Italia, 2014.
Reference Bibliography
Material given by the professor via the e-learning page of the course, including lecture slidesType of delivery of the course
frontal lessons, exercisesAttendance
Recommended frequencyType of evaluation
oral / written teacher profile teaching materials
Linear equations systems. Matrix associated with a linear system. Equivalent systems. Natural, integer, rational numbers, real numbers and their property. Recall of set theory: inclusion of sets, difference between sets.
2. Matrices and sets
Matrices with real coefficients. Square, triangular, diagonal matrices. Transpose of a matrix and symmetric matrices. Recall of set theory: union and intersection of sets.
3. The vector space of the matrices
Addition between matrices and its properties. Multiplying a scalar for a matrix and its properties.
4. Product between matrices
Product between matrices with compatible dimensions. Properties of the product: associative property and distributive property. Examples showing that product between matrices does not satisfy the commutative property and the simplification property. Matrices and linear systems.
5. Determinants
Definition by induction of the determinant when using the first-row development. Determinant property: development according to any row or column, determinant of the transposed matrix, determinant of a triangular matrix. Binet theorem.
6. Reverse matrix
Unit matrix. Reverse matrix. Inverse property. Cramer's theorem.
7. Rank of a matrix
Definition. Property of the rank. Minors of a matrix. Theorem of Kronecker.
8. Linear equation systems
Definitions. Rouché-Capelli theorem. Rouché-Capelli method for solving a linear system.
9. Gauss method
10. Applications of Gauss method
Basic operations. Calculation of the determinant. Calculation of the rank.
11. Geometric vectors
Plan vectors. Addition between vectors. Product of a vector for a scalar. Space vectors. Lines and planes for the origin. Average point.
12. Linear combinations of geometric vectors
Linear combinations. Linearly dependent and independent vectors. Characterization of linearly independent vectors in V2(O) and V3(O).
13. Vector spaces on the real numbers
Definition of vector spaces. Examples of vector spaces. Basic properties of vector spaces.
14. Vector subspaces
Definition of vector spaces. Subspaces of V2(O) and V3(O).
15. Generators of vector spaces
Linear combinations and generators.
16. Linear dependency and independency
17. Basis of vector spaces
Basis. Dimension. Dimension of the set of solutions of a homogeneous system. Dimension of subspaces. Calculation of dimensions and basis.
18. Intersection and sum of subspaces
Intersection of vector subspaces. Sum of vector subspaces. Grassmann's formula.
19. Affine subspaces
The lines of the plane and of the space. Space planes. Affine subspaces. Set of system solutions.
20. Homomorphisms
Homomorphisms between vector spaces. Matrix associated with a homomorphism. Homomorphism associated with a matrix.
21. Image
Property of the image of a homomorphism. Calculation of the image of a homomorphism. Condition of surjectivity of a homomorphism.
22. Kernel
Property of the kernel of a homomorphism. Calculation of the kernel of a homomorphism. Injectivity condition of a homomorphism.
23. Endomorphisms
Matrix associated with an endomorphism. Change of basis.
24. Eigenvalues and eigenvectors
Definitions and basic properties. Eigenspaces. Characteristic polynomial. Diagonalizable matrices.
25. Diagonalization
Diagonalizability conditions. Diagonalization procedure.
26. Graph optimization:
Maximum flow. Shortest path. Minimum spanning tree.
* This book is available for free at the following link: http://www.dmmm.uniroma1.it/accascinamonti/geogest/Geometria.pdf
Caramia, Giordani, Guerriero, Musmanno, Pacciarelli, "Ricerca Operativa", Isedi, Italia, 2014.
Programme
1. Linear equations and numbersLinear equations systems. Matrix associated with a linear system. Equivalent systems. Natural, integer, rational numbers, real numbers and their property. Recall of set theory: inclusion of sets, difference between sets.
2. Matrices and sets
Matrices with real coefficients. Square, triangular, diagonal matrices. Transpose of a matrix and symmetric matrices. Recall of set theory: union and intersection of sets.
3. The vector space of the matrices
Addition between matrices and its properties. Multiplying a scalar for a matrix and its properties.
4. Product between matrices
Product between matrices with compatible dimensions. Properties of the product: associative property and distributive property. Examples showing that product between matrices does not satisfy the commutative property and the simplification property. Matrices and linear systems.
5. Determinants
Definition by induction of the determinant when using the first-row development. Determinant property: development according to any row or column, determinant of the transposed matrix, determinant of a triangular matrix. Binet theorem.
6. Reverse matrix
Unit matrix. Reverse matrix. Inverse property. Cramer's theorem.
7. Rank of a matrix
Definition. Property of the rank. Minors of a matrix. Theorem of Kronecker.
8. Linear equation systems
Definitions. Rouché-Capelli theorem. Rouché-Capelli method for solving a linear system.
9. Gauss method
10. Applications of Gauss method
Basic operations. Calculation of the determinant. Calculation of the rank.
11. Geometric vectors
Plan vectors. Addition between vectors. Product of a vector for a scalar. Space vectors. Lines and planes for the origin. Average point.
12. Linear combinations of geometric vectors
Linear combinations. Linearly dependent and independent vectors. Characterization of linearly independent vectors in V2(O) and V3(O).
13. Vector spaces on the real numbers
Definition of vector spaces. Examples of vector spaces. Basic properties of vector spaces.
14. Vector subspaces
Definition of vector spaces. Subspaces of V2(O) and V3(O).
15. Generators of vector spaces
Linear combinations and generators.
16. Linear dependency and independency
17. Basis of vector spaces
Basis. Dimension. Dimension of the set of solutions of a homogeneous system. Dimension of subspaces. Calculation of dimensions and basis.
18. Intersection and sum of subspaces
Intersection of vector subspaces. Sum of vector subspaces. Grassmann's formula.
19. Affine subspaces
The lines of the plane and of the space. Space planes. Affine subspaces. Set of system solutions.
20. Homomorphisms
Homomorphisms between vector spaces. Matrix associated with a homomorphism. Homomorphism associated with a matrix.
21. Image
Property of the image of a homomorphism. Calculation of the image of a homomorphism. Condition of surjectivity of a homomorphism.
22. Kernel
Property of the kernel of a homomorphism. Calculation of the kernel of a homomorphism. Injectivity condition of a homomorphism.
23. Endomorphisms
Matrix associated with an endomorphism. Change of basis.
24. Eigenvalues and eigenvectors
Definitions and basic properties. Eigenspaces. Characteristic polynomial. Diagonalizable matrices.
25. Diagonalization
Diagonalizability conditions. Diagonalization procedure.
26. Graph optimization:
Maximum flow. Shortest path. Minimum spanning tree.
Core Documentation
G. Accascina and V. Monti, Geometry** This book is available for free at the following link: http://www.dmmm.uniroma1.it/accascinamonti/geogest/Geometria.pdf
Caramia, Giordani, Guerriero, Musmanno, Pacciarelli, "Ricerca Operativa", Isedi, Italia, 2014.
Reference Bibliography
Material given by the professor via the e-learning page of the course, including lecture slidesType of delivery of the course
frontal lessons, exercisesAttendance
Recommended frequencyType of evaluation
oral / written