The course aims to provide an introduction to those aspects of linear and discrete mathematics needed in science and engineering.
Curriculum
Canali
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.
* This book is available for free at the following link: http://www.dmmm.uniroma1.it/accascinamonti/geogest/Geometria.pdf
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.
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
Reference Bibliography
information available on the moodle page of this courseType of delivery of the course
Frontal teaching lessons. Classroom exercises.Attendance
Frequency suggested but not mandatory.Type of evaluation
Learning verification takes place through a written test lasting a few hours, plus an optional oral test or an oral test requested by the teacher. The written exam is organized through exercises and theoretical questions, aimed at verifying the level of effective understanding of the course contents. Some exam exercises (and intermediate tests) of previous years are available on the moodle page of the course, including a detailed explanation on how to solve of these exercises. Self-assessment tests and classroom exercises are scheduled during the course. "In the COVID-19 emergency period, the profit examination will be carried out in accordance with article 1 of the Rectoral Decree no. 703 of May 5th, 2020" teacher profile teaching materials
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.
"Geometria"
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
Lessons and exam simulations.Attendance
Attending class is recommended.Type of evaluation
During the COVID-19 emergency the exam will follow art. 1 of the D.R. n°703 of 5th may 2020Canali
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.
* This book is available for free at the following link: http://www.dmmm.uniroma1.it/accascinamonti/geogest/Geometria.pdf
Mutuazione: 20810098-2 GEOMETRIA E COMBINATORIA II MODULO in Ingegneria informatica L-8 CANALE 1 D'ARIANO ANDREA
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.
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
Type of delivery of the course
Frontal teaching lessons. Classroom exercises.Attendance
Frequency suggested but not mandatory.Type of evaluation
Learning verification takes place through a written test lasting a few hours, plus an optional oral test or an oral test requested by the teacher. The written exam is organized through exercises and theoretical questions, aimed at verifying the level of effective understanding of the course contents. Some exam exercises (and intermediate tests) of previous years are available on the moodle page of the course, including a detailed explanation on how to solve of these exercises. Self-assessment tests and classroom exercises are scheduled during the course. "In the COVID-19 emergency period, the profit examination will be carried out in accordance with article 1 of the Rectoral Decree no. 703 of May 5th, 2020" teacher profile teaching materials
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.
"Geometria"
Mutuazione: 20810098-2 GEOMETRIA E COMBINATORIA II MODULO in Ingegneria informatica L-8 CANALE 2 SAMA' MARCELLA
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
Lessons and exam simulations.Attendance
Attending class is recommended.Type of evaluation
During the COVID-19 emergency the exam will follow art. 1 of the D.R. n°703 of 5th may 2020