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

Material given by the professor via the e-learning page of the course, including lecture slidesType of delivery of the course

frontal lessons, exercisesType of evaluation

intermediate / final, oral / written 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"

Reference Bibliography

Giulia Maria Piacentini Cattaneo "Matematica discreta" Edito da Zanichelli.Type of delivery of the course

Teaching includes lectures, exercises with the teacher and self-assessment tests for students. Attendance is not mandatory but is strongly recommended.Attendance

Teaching includes lectures, exercises with the teacher and self-assessment tests for students. Attendance is not mandatory but is strongly recommended.Type of evaluation

intermediate / final, oral / written 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 evaluation

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

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 evaluation

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