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). For example, the student is able to symbolically deal with a linear system of equations and to compute its solutions: these solutions can correspond to the intersection of two lines, to the eigenvectors of a matrix or to the circles passing through two points. 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 solutions 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 problems, Stabilty 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 method. 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.

Modalità di esame: Test informatizzato in laboratorio; Prova scritta (in aula); Prova orale facoltativa;

Exam: Computer lab-based test; Written test; Optional oral exam;

... The exam will check the theoretical knowledge of the student on the course syllabus and it will also check the ability of the student of applying the theoretical knowledge in practical situation. The exam will also check the ability of the student of identifying and applying the best, theoretical or numerical, method to treat some basic problems and in interpreting the obtained results. Thus, the exam will deal with the theoretical and the applied aspects of the course and will be performed through a computer based test, a written test and an optional oral exam (see below). The computer assisted test consists of 8 multiple choice questions on theoretical and practical aspects dealing with the application of the numerical methods met during the course; to answer to most of the questions it is necessary to use the software MATLAB. The computer assisted test has a duration of 45 min and will give up to 10 points. There is a threshold of 3 points, if the student gets no more than 3 points the student fails the exam. The written test has a duration of 60 min and consists of 8 multiple choice and an exercise. The multiple choice questions give up to 16 points and the exercises give up to 7 points. Both the multiple choice questions, both the exercise deal with theoretical and practical aspects of linear algebra and geometry. There is a threshold of 6 points for the multiple choices and of 2 points for the exercise. If the student gets less than 6 points in the multiple choices, then the student fails the exam. If the student gets less than 2 points in the exercise, then the student fails the exam. The teacher can ask the student to take an oral test (only in the case that the student did not fail the computer assisted test or the written test and if the student has a total of at least 18 points). The final mark is the sum of the points the student obtained in the computer assisted test and in the written test; the total can be up to 30L. In the case of an oral test this will be taken into account int he final grade. During the computer assisted test and the written test it is forbidden to use books, notes, electronic devices besides the PC of the LAIB, and any not authorised material.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.