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 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 two computer based tests, a written test and an optional oral exam (see below). The first 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. This computer assisted test has a duration of 45 min and will give up to 10 points; each wrong answer has a penality of 20%; there is no penality for not answered questions. The second computer assisted test consists of 8 multiple choice questions on theoretical and practical aspects of linear algebra and geometry. This computer assisted test has a duration of 45 min and will give up to 16 points; each wrong answer has a penality of 20%; there is no penality for not answered questions. If the student gets no more than 3 points in the first test or no more than 6 points in the second test, then the student fails the exam. The written test has a duration of 30 min and consists of open questions on the theoretical and practical aspects of linear algebra and geometry. If the student does not get more than 2 points, 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 tests 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 tests 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 in the final grade. During the computer assisted tests and the written test it is forbidden to use books, notes, electronic devices besides the PC of the LAIB, and any not authorised material.

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.

Exam: Optional oral exam; Computer-based written test using the PoliTo platform;

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 three computer based tests, and an optional oral exam (see below). The first computer assisted test consists of 8 closed or open questions on theoretical and practical aspects dealing with the application of the numerical methods met during the course; during the exam it will not be possible to use the software MATLAB but it is necessary to know the syntax of its commands. This computer assisted test has a duration of 45 min and will give up to 10 points; each wrong answer has a penality of 20%; there is no penality for not answered questions. The second computer assisted test consists of 8 closed or open questions on theoretical and practical aspects of linear algebra and geometry. This computer assisted test has a duration of 45 min and will give up to 16 points; each wrong answer has a penality of 20%; there is no penality for not answered questions. If the student gets no more than 3 points in the first test or no more than 6 points in the second test, then the student fails the exam. The third computer assisted test has a duration of 30 min and consists of open or closed questions on the theoretical and practical aspects of linear algebra and geometry. If the student does not get more than 2 points, 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 tests 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 tests; the total can be up to 30L. In the case of an oral test this will be taken into account in the final grade. 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; Written test; Computer-based written test using the PoliTo platform;

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 two computer based tests, a written test (only onsite) or a computer assisted test (only online) and an optional oral exam (see below). The first computer assisted test consists of 8 closed or open 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, possibly using VLaib. This computer assisted test has a duration of 45 min and will give up to 10 points; each wrong answer has a penality of 20%; there is no penality for not answered questions. The second computer assisted test consists of 8 closed or open questions on theoretical and practical aspects of linear algebra and geometry. This computer assisted test has a duration of 45 min and will give up to 16 points; each wrong answer has a penality of 20%; there is no penality for not answered questions. If the student gets no more than 3 points in the first test or no more than 6 points in the second test, then the student fails the exam. The written test (onsite) and the computer assisted test (online) have a duration of 30 min each and consist of open and closed questions on the theoretical and practical aspects of linear algebra and geometry. If the student does not get more than 2 points, 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 tests 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 tests 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 in the final grade. During the computer assisted tests and the written test it is forbidden to use books, notes, electronic devices besides the PC of the LAIB, and any not authorised material. The online test will be performed using the platform Exam with a proctoring system (Respondus).