Course Topics
This course provides a comprehensive introduction to fundamental concepts of linear algebra and discrete mathematics, with a strong emphasis on applications to computer science and digital business. It develops both theoretical foundations and problem-solving skills, combining lectures with practical exercises. Linear Algebra Vectors: addition, scalar multiplication, linear combinations, representations, dot product, orthogonality, vector lengths and angles, unit vectors, and vector inequalities (Cauchy–Schwarz, triangle inequalities). Matrices: notation, basic operations (addition, scalar multiplication, transpose, conjugate transpose), matrix multiplication, identity and inverse matrices, powers of matrices, properties of matrix operations. Linear Systems: formulation, solution methods, inverse matrices, independence and dependence, singular systems, practical examples (Google PageRank, electrical circuits). Gaussian Elimination: elimination process, back substitution, equivalent systems, operation counts, common breakdowns (no solution, infinite solutions, zero pivots). LU Factorization and Gauss–Jordan Method: triangular factorization, systematic solution of systems. Rectangular Systems and Rank: echelon forms, reduced row echelon form, rank factorization, column relations, and consistency criteria. Geometric Interpretations: planes, algebra of planes, consistency and meaning of rank. Discrete Mathematics Graphs: basic definitions and terminology, graph representations (adjacency matrix, directed/undirected graphs), applications to networks, the World Wide Web, and knowledge representation. Graph Properties: simple graphs, complete and bipartite graphs, subgraphs, degrees, and vertex properties. Walks on Graphs: Euler trails and circuits, connected components, adjacency matrices, counting walks of fixed length. Graph Centrality: degree, closeness, and betweenness centrality, and their interpretation in network analysis. Logic of Compound Statements: propositions, logical operators, truth tables, negations, conjunction, disjunction, exclusive or, inequalities, and order of operations. Logical Equivalence: De Morgan’s laws, tautologies, contradictions, simplifying statement forms, and equivalence transformations. Validity of Arguments: conditional statements, contrapositives, converse/inverse, necessary and sufficient conditions, Modus Ponens, Modus Tollens, proof by cases, contradiction rule. Mathematical Induction and Recursion: principle of induction, examples of recursive definitions and proofs. Additional Background Topics Review of complex numbers, trigonometry, and polynomials as needed. Sets, functions, and basic counting principles. The course integrates theory with practical applications, encouraging students to connect abstract mathematical concepts with real-world computational problems. Students will gain experience in formal reasoning, problem-solving, and the use of mathematics as a tool for computer science and data-driven decision-making.

Teaching format
This course will be delivered through a combination of formal lectures and exercises.

Educational objectives
Type of course: “di base” for L-31 Scientific area: “Formazione matematica-fisica” for L-31 The aim of this course is to present a rather comprehensive treatment of linear algebra and discrete mathematics, giving a general overview of the field, giving a general overview of the field. It covers vector, matrix and numbers theory, sets, functions and graphs to some degree of mathematical logic and rigour, emphasizing topics that are in support of computer science. The course also provides practice in using the tools of mathematics to solve problems and to make judgements autonomously. Knowledge and understanding: • D1.1 - Possess basic knowledge of mathematical analysis, algebra, numerical calculation and optimisation methods which support computer science and advanced economics. Applying knowledge and understanding: • D2.1 - Ability to use mathematics and statistical data analysis tools to solve computational problems. Learning skills • D5.1 - Learning ability to undertake further studies with a high degree of autonomy.

Assessment
Written exam, consisting of a set of verification questions, transfer of knowledge questions and exercises. The aim of the assessment is to check to which degree students have mastered the following learning outcomes: 1) knowledge and understanding, 2) applying knowledge and understanding, 3) making judgment. The same rules apply to both attending and non-attending students.

Evaluation criteria
Final Written Exam, 100% covering the full program. Written exam questions will be evaluated in terms of correctness, clarity, quality of argumentation, problem solving ability. The same rules apply to both attending and non-attending students.