The course has two main goals. The first one is to introduce the main topics of linear algebra and geometry, training the student to follow logical deductive arguments and to use the proper formal language. The second goal is to give to the students the main concepts of some basic numerical methods of linear algebra and of their implementation in MATLAB, which is by now widely used in engineering. The course will show how theoretic, symbolic and numerical aspects interact with each other.

This course wants to develop the student's ability to understand logical arguments stressing the role of the hypothesis, for example by building examples and counterexamples. The student acquires tools and techniques to work with geometrical objects (vectors in the plane and i the space, lines, planes, conics and quadrics) and with algebraic objects (linear systems of equations, matrices, polynomials, eigenvalues, eigenvectors, vector spaces and their transformations). The student will also acquire the necessary knowledge to numerically solve, also using a computer, some basic problems in linear algebra when the “pencil and paper” method is not feasible (for example, computing the solution of a large linear system of equations and computing the eigenvalues of a matrix). In particular, the student learns to identify an algebraic/geometric object, to recognize its theoretical properties and to choose the most fit algebraic/geometrical method to deal with the object. The implementation and the application of the numerical methods is done using the software MATLAB of which the student learns the basic aspects.

A working knowledge of the mathematical tools presented in the first semester. In particular, a basic knowledge of real and complex numbers, solving equations and inequalities of degree one or two, differential and integral calculus as given in Mathematical Analysis I, as well as the main syntactic constructs used in computer programming, taught in the course of Computer Sciences.

• Vector in 2-space and in 3-space and their operations. Dot product, cross product and box product. Lines and planes in 3-space. Orthogonal projections. • Matrices and their operations. Strongly reduced matrices. Matrix form of linear systems of equations and their solutions with geometrical applications. Matrix equations and inverse of a matrix. Determinants. • Vector spaces: definition, examples and applications. Sub-vector spaces and main operations with them. • Linear combination and linearly dependent vectors. How to extract linearly independent vectors from a set. Bases of a vector space. Dimension of a vector space. Dimension of finitely generated subspace. • Space of polynomials. Grassmann's relation. • Linear maps. Image of a linear map. Injective and surjective linear maps. Isomorphisms. • Matrix of a linear map. Endomorphism and square matrices. • Eigenvalues and eigenvectors. Eigenspaces of matrix endomorphisms. Characteristic polynomial of an endomorphism. Diagonalization of and endomorphism. • Orthonormal bases, orthonormal matrices. Gram-Schmidt's algorithm. Diagonalization of real symmetric matrices using orthogonal matrices. Quadratic forms and the sign that they can take in a point. • Metric problems: distance between two points, two lines, and a point and a line. • Quadratic geometry: conic curves, and spheres. Non-degenerate quadrics in canonical form. Recognising a quadric surface. •Machine arithmetic, machine numbers, rounding error. Conditioning of a numerical problem, stability of an algorithm. •Approximation of functions and data : polynomial interpolation and piecewise polynomial interpolation (spline). Main results about convergence. •Linear systems: conditioning and numerical direct methods. Matrix factorizations PA=LU, Choleski, QR and their main applications. •Eigenvalues of matrices: conditioning and numerical methods (powers, inverse power, QR (basics notions)). Singular values decomposition of matrices and its main applications.

Lectures (60 hours), practical sessions in the classroom (30 hours), and computer assisted practical sessions (10 hours).

A. Quarteroni, F. Saleri, P. Gervasio, Scientific Computing with MATLAB and OCTAVE, Springer, 2014. G. Strang, Introduction to Linear Algebra, Wellesley-Cambridge Press, 2016. E. Carlini, LAG: the written exam, v1 and v2 CLUT 2019. E. Carlini, 50 multiple choices in Geometry, CELID 2011.

Exam: Computer-based written test with open-ended questions or multiple-choice questions using the Exam platform and proctoring tools (Respondus);

The exam will check the theoretical knowledge of the material of the course and the ability of applying the theory in a practical context. It will also be checked the ability of the student of choosing and applying a numerical method depending on the explicit example. Thus the exam will focus on both theoretical and applied aspects of the course and will be performed using a PC assisted test using both open and closed questions. The test consists of 14 questions of which 10 are on the theoretical part and 4 are on the applied part. The test has a duration of 75 minutes and has a maximum score of 33 points; each question has the same value. Each multiple choice question has 4 alternatives of which only one is correct; each wrong answer has a penalty of 15% of its value and a not given answer gives 0 points. The teacher can ask the student to take an oral test (only in the case that the student of at least 18 points) to further test the theoretical knowledge of the student. During the test it is forbidden to use books, notes and not authorised electronic devices. The above described test will be performed using the platform Exam with a proctoring system (Respondus).

Exam: Computer lab-based test; Computer-based written test with open-ended questions or multiple-choice questions using the Exam platform and proctoring tools (Respondus);

The above described test will be performed either on-site or on-line. The on-site exam will be performed in a LAIB using the platform Exam. The on-line exam will be performed using the platform Exam with proctoring tools (Respondus).