LINEAR ALGEBRA

The course Algebra Lineare is one of the fundamental educational activities of the three-year degree program in Ingegneria Fisica and allows the student to approach mathematical problems with a consistent use of modern mathematical language.
The aim of the course is to provide knowledge and skills related to the theoretical and applicative foundations of geometry and linear algebra.
At the end of the course, the student will have acquired the necessary foundations to tackle the mathematical models developed in other courses of the degree program.

Knowledge and Understanding
At the end of the course, students will have acquired fundamental concepts of Linear Algebra, with a particular focus on the concept of linearity. They will understand the principles of vector calculus and develop a solid grasp of matrices, vector spaces, and linear applications. Additionally, they will be familiar with the definitions and the geometric/algebraic symbolism used in the discipline.

Ability to Apply Knowledge and Understanding
Students who have completed the course will be able to reason logically and apply mathematical symbolism appropriately. They will have the ability to formulate and implement strategies for solving problems in Linear Algebra and will recognize the role of mathematics in other scientific disciplines.

Autonomy of Judgment
Students will be able to critically assess the logical consistency of the results obtained, both in theoretical contexts and in practical mathematical problems. They will develop the ability to identify errors through a critical analysis of the applied methods and a verification of the results. Furthermore, they will be able to evaluate and compare different approaches to solving mathematical problems.

Communication Skills
Students will develop the ability to effectively communicate the concepts learned, using appropriate mathematical terminology, both orally and in written form. They will also be able to interact constructively with peers and instructors, asking relevant questions and proposing alternative problem-solving methods.

Learning Ability
The knowledge acquired will enable students to take notes effectively, selecting and organizing information based on relevance and priority. They will develop the ability to consult textbooks recommended by the instructor and identify alternative references when needed. Additionally, they will be able to apply the concepts learned to correctly approach and solve mathematical problems.

Knowledge of high-school level mathematics

- Complex numbers: definition, representations of complex numbers, fundamental operations, Euler's formula, Fundamental Theorem of Algebra.

- Vectors in the plane and in the space: fundamental operations, scalar and vectorial product, linear dependence and independence (geometric meaning).
- Analytical Geometry: Lines and Planes in Space.

- Matrices: definition, sum and product between matrices, transposed matrix. Determinant of a square matrix, property of the determinant and Sarrus rule. Inverse matrix and rank of a matrix, Gaussian elimination method.
- Linear systems: resolution methods and geometric meaning, Cramer's and Rouchè Capelli's theorems.

- Vector spaces: definition in real and complex fields, basis and size of a vector space. Orthonormal bases. Examples of vector spaces (polynomials, matrices and functions). Vector subspaces.
- Linear Applications: definition, core and image of a linear application, matrix associated with a linear application between spaces of finite dimension. Change of basis, invertible linear applications.

- Eigenvalues ​​and eigenvectors: definition and geometric meaning. Diagonalizable matrices, algebraic and geometric multiplicity of an eigenvalue and geometric meaning, definition of autospace. Diagonalization theorem. Spectral theorem.

Algebra Lineare. M. Abate,McGraw-Hill
Algebra Lineare e Geometria, F. Bottacin, Società Editrice Esculapio
Analisi matematica 1. Con elementi di algebra lineare, M. Bramanti, C. Pagani, S. Salsa, Zanichelli

The student must demonstrate their understanding of the topics covered in the course through a written test, which includes exercises and theoretical questions on all the subjects studied during the lectures. The evaluation will consider the accuracy of the solutions, the clarity and completeness of the justifications, the proper use of scientific language, and the ability to apply the tools of Linear Algebra.

written

18-21: Basic understanding with gaps; can perform fundamental operations but struggles with abstract concepts and theoretical justifications.

22-23: Competent in basic methods but has difficulty with deeper theoretical aspects.

24-27: Good grasp of most topics; handles vector spaces, linear systems, and eigenvalues well but may lack fluency in proofs and abstract reasoning.

28-29: Strong theoretical and computational understanding; rigorously justifies results and connects algebraic and geometric interpretations.

30: Mastery of all topics; provides clear, structured justifications and demonstrates deep understanding of eigenvalues, vector spaces, and transformations.

30 cum laude: Exceptional insight and critical thinking; extends topics beyond the syllabus with elegant proofs and original reasoning.

Theory and exercises lectures.

Accommodation and support services for students with disabilities and students with specific learning impairments:
Ca’ Foscari abides by Italian Law (Law 17/1999; Law 170/2010) regarding support services and accommodation available to students with disabilities. This includes students with mobility, visual, hearing and other disabilities (Law 17/1999), and specific learning impairments (Law 170/2010). In the case of disability or impairment that requires accommodations (i.e., alternate testing, readers, note takers or interpreters) please contact the Disability and Accessibility Offices in Student Services: disabilita@unive.it.