LINEAR ALGEBRA

The teaching ALGEBRA LINEARE is one of the basic activities of the three-year degree course in Ingegneria Fisica, and allow the student to face a mathematical problem with a consistent use of the current mathematical language.
The specific goal of this teaching is the formation of the knowledge and the skills concerning the theoretical and applicative basis of the geometry and linear algebra. The teaching will form the basis to deal with the mathematical models developed in the other courses of the Degree.

1. Knowledge and understanding
i) To know the basic concepts of aLinear Albegra, and in particular the concept of linearity.
ii) To know how to use vectorial calculus, to understand the concepts of matrices, vector spaces and linear applications.
iii) To know the definitions and the geometric/algebraic symbolism.
2. Ability to apply knowledge and understanding.
i) To know how to reason in a logical way and how to use mathematical symbolism in an appropriate way.
ii) To understand the linear albegra and to know how to set up a strategy to solve problems.
iii) To know how to recognize the role of mathematics in other sciences.
3. Ability to judge
i) Being able to evaluate the logical consistency of the results obtained, both in the theoretical field than in the case of concrete mathematical problems.
ii) Being able to recognize errors through a critical analysis of the method applied and through a control of the results obtained.
iii) To evalute the possibility of different approaches when solving mathamtical problems.
4. Communication skills
i) To know how to communicate what have been learned by using an appropriate terminology, also in written form.
ii) To know how to interact with the teacher and with the classmates in a respectful and constructive way, by asking coherent questions and by proposing other ways to solve a problem.
5. Learning skills
i) To know how to take notes in an effective way, selecting and collecting information according to their importance and priority.
ii) To know how to consult the books given by the teacher, and to know how to identify alternative references, also through the interaction with the teacher.
iii) Being able to exploit the concepts learned to correctly perform a mathematical problem.

Knowledge of mathematics for high-school level

The main topics of the course are the following:

- 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
Appunti di Lagebra Lineare, on moodle.

The exam consists of a written test with exercises and theoretical questions concerning all the topics studied during the classes. In the written test, the correctness of the exposure, the clarity and completeness of the justifications, the knowledge of the scientific language and the ability to use the tools of linear algebra will be evaluated.
The written test will last between two and four hours.

Lectures: theory and exercises.
Educational material will be found in the "moodle" platform.

Italian

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 supportservices 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.

written