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. •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.

Lecture slides; Text book; Practice book; Exercises; Exercise with solutions ; Lab exercises; Lab exercises with solutions; Video lectures (previous years); Simulation tools; Self-assessment tools;

You can take this exam before attending the course

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

The exam will check the student's theoretical knowledge of the course syllabus and the ability to apply the theoretical knowledge in practical situations. The exam will also check the ability of the student in 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 applied aspects of the course. The exam includes: 1) a first part consisting of a computerized test; 2) a second part consisting of a written test. The computer-assisted test consists of 8 closed-answer or open-answer questions concerning the theoretical and practical aspects dealing with the application of the numerical methods discussed during the lectures; to answer most of the questions it is necessary to use the software MATLAB. This test, lasting 45 minutes, scores up to 10 points; each wrong answer has a penalty of 15% for closed questions; there is no penalty for not answering questions. This part of the exam is passed if the student obtains a score greater than or equal to 4.25 points. Passing this part is necessary to be admitted to the second part. The written test, lasting 60 minutes, consists of 8 closed-answer questions and an exercise with open-answer questions concerning the theoretical and practical aspects of linear algebra and geometry. Each correct answer to a closed-answer question scores 2 points while the exercise allows the student to achieve up to 7 points. There will be no penalty for an incorrect answer. A necessary condition for passing the exam is that the student obtains a score greater than or equal to 7 points for closed-answer questions and greater than or equal to 2 points for the exercise. The teacher, at their discretion, may request an oral test (only if the student has reached the thresholds of the previous tests and has achieved a total score of at least 18 points) which is intended to further ascertain the learning of the theory, constituting an additional element of evaluation. During the semester, two mid-semester tests are carried out. The tests consist of integrated exercises (numerical and algebraic-geometric part) which will concern both theoretical and practical aspects, and whose resolution will also require the use of numerical methods. Each mid-semester test allows the student to achieve up to 3 points, therefore a maximum total score of 6. This score, only in the summer and autumn exam sessions of the academic year in which it was achieved, contributes to the final grade as follows: a) 1/3 will contribute to the score of the first part which, in any case, cannot exceed 10, therefore First part score = minimum {10, computer-assisted test score + 1/3 of the score acquired in mid-semester tests}, and the score of the first part thus calculated must be greater than or equal to 4.25 points to pass the first part of the exam; b) 2/3 will contribute to the score of the written test, limited to the part relating to closed-answer questions. In any case, the score of this part cannot exceed 16, therefore Second part score = minimum {16, closed-answer questions score + 2/3 of the score acquired in the mid-semester tests} + score acquired with the open-answer exercise, and the sum of the score obtained with the closed-answer questions and the contribution of the mid-semester tests must be greater than or equal to 7 points to pass the exam. The final grade is given by the sum of the scores achieved in the first and second parts of the exam. A mark of at least 18 implies passing the exam. A score higher than 31 implies the assignment of honours. In case the oral test is carried out, the final grade will be established taking into account both the score already achieved in the tests mentioned above, and the outcome of the oral test. During the tests, the use of books, notes, electronic devices other than the laboratory PC, and other unauthorized material is prohibited.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.