Università Roma Tre

Elements of set theory. Maps between sets: invective, surjective, bijective maps.
Elements of propositional logic, truth tables. Equivalence and order relations.
Combinatorics. Binomial coefficients and binomial theorem. Permutations. The Integers: divisibility, GCD and Euclidean algorithm, Bézout identity, linear congruences.
Baiscs of algebraic structures: permutation groups, abstract groups, polynomials and finite fields.
Elements of graph theory.Lattices and Boolean algebras

Giulia Maria Piacentini Cattaneo
Matematica discreta e applicazioni
Zanichelli 2008

Sets. Operations between sets. Applications between sets. Permutations. Relations. Equivalence relations. Order relations. Lattices. Algebraic structures. Group of permutations of n elements. Vector spaces. Linear dependence and independence of vectors. Basis. Dimension. Subspaces. Subspace sum. Subspace intersection. Grassmann’s relation. Matrices. Determinants. Inverse matrix. Reduction of a matrix into treads. The rank of a matrix. Linear systems. Theorem of Rouche-Capelli. Cramer's theorem. Method of Gaussian elimination.

F.J. Leon Trujillo, P. Mercuri. Elementi di Algebra Lineare, Ed. Efesto
F.J. Leon Trujillo, P. Mercuri. Elementi di Geometria Affine ed Euclidea, Ed. Efesto
F.J. Leon Trujillo. Esercizi di Algebra Lineare e Geometria, Ed. Efesto

1. Linear equations and numbers
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.

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

Linear applications. Kernel. Image. Linear maps and linear dependence. Linear maps and matrices. Change of basis. Similar matrices. Diagonalization of an endomorphism. Eigenvalues. Eigenvectors. Eigenspaces. Characteristic polynomial. Scalar products. Euclidean vector spaces. Diagonalization of symmetric endomorphisms. Elements of affine and Euclidean geometry of the plane and space.

F.J. Leon Trujillo, P. Mercuri. Elementi di Algebra Lineare, Ed. Efesto
F.J. Leon Trujillo, P. Mercuri. Elementi di Geometria Affine ed Euclidea, Ed. Efesto
F.J. Leon Trujillo. Esercizi di Algebra Lineare e Geometria, Ed. Efesto