


Politecnico di Torino  
Academic Year 2017/18  
03KXTLZ, 03KXTJM, 03KXTLI, 03KXTLM, 03KXTLN, 03KXTLP, 03KXTLS, 03KXTLX, 03KXTMA, 03KXTMB, 03KXTMC, 03KXTMH, 03KXTMK, 03KXTMN, 03KXTMO, 03KXTMQ, 03KXTNX, 03KXTOA, 03KXTOD, 03KXTPC, 03KXTPI, 03KXTPL Linear algebra and geometry 

1st degree and Bachelorlevel of the Bologna process in Aerospace Engineering  Torino 1st degree and Bachelorlevel of the Bologna process in Mechanical Engineering  Torino 1st degree and Bachelorlevel of the Bologna process in Automotive Engineering  Torino Espandi... 





Subject fundamentals
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 MATLAB programming language, which is by now widely used in engineering. The student will learn to tackle and solve some linear algebra problems, that generally occur as intermediate steps in the solution of more complex mathematical models for engineering problems and that cannot be solved by analytical means.

Expected learning outcomes
The student will be able to follow proofs, to build examples and counterexamples and to use the basic properties of analytic geometry and of vector spaces. The student will be able to work with matrices and to solve systems of linear equations. Moreover, given a mathematical problem of linear algebra (e.g. solution of a linear system or computation of eigenvalues of a matrix), the student will be able to identify and to apply (using MATLAB) the most efficient numerical method among the presented ones, which gives an approximation of the solution with the best possible accuracy and with the lowest computational cost.

Prerequisites / Assumed knowledge
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 1, as well as the main syntactic constructs used in computer programming, taught in the course of Computer Sciences.

Contents
• Vector in 2space and in 3space and their operations. Dot product, cross product and box product. Lines and planes in 3space. 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. Subvector spaces and main operations with them. • Linear combination and linearly dependent vectors. How to extract linearly independent vectors from a set. Bases of a vectors 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. GramSchmidt'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. Nondegenerate quadrics in canonical form. Recognising a quadric surface. • General concepts on numerical problems and algorithms. Machine numbers, rounding error. • Approximation of functions and numerical data. Polynomial interpolation: Lagrange and Newton representations. Choice of interpolation points and convergence properties. Piecewise polynomial interpolation: splines. Overdeterminated and underdeterminated linear systems. • Numerical solution of linear systems. Matrix norms and condition number of a linear system. Substitution technique of triangular linear systems. Gauss elimination method. Partial pivoting. PA=LU factorization and its applications. Choleski factorization and its applications. QR factorization. Least squares method. • Numerical computation of eigenvalues. Power method and inverse power method. QR method. Singular value decomposition and its applications. 
Delivery modes
Exercises will cover the topics of the lectures. Some will be carried out by the teacher at the blackboard, others will actively involve the students. Moreover, computer room activities are also scheduled, during which the numerical algorithms presented in the lectures will be implemented by using the MATLAB software. The application of these algorithms will allow to study in a deeper way the theoretical aspects and the properties of the presented numerical methods, and to perform a critical analysis of the numerical results.

Texts, readings, handouts and other learning resources
In English (suggested in the case the provided lecture notes are not considered enough)
L. Robbiano, Linear Algbera for Everyone, Springer Science & Business Media, 09 mag 2011 – 218 pp. E.Carlini, 50quiz in Geometry, Celid 2011. In Italian (suggeriti) L. Gatto, Lezioni di Algebra lineare e Geometria, Clut 2013. S.Greco, P. Valabrega, Lezioni di Geometria, Vol. 1 Algebra lineare, Vol. 2 Geometria Analitica, Ed. Levrotto e Bella, Torino 2009. G. Casnati, M.L. Spreafico, Allenamenti di Geometria, Ed. Esculapio, Bologna 2013. J. Cordovez, Chissŕ chi lo sa?, Clut 2013. G. Monegato, Metodi e algoritmi per il Calcolo Numerico, CLUT 2008. L. Scuderi, Laboratorio di Calcolo Numerico. Esercizi di Calcolo Numerico risolti con Matlab, CLUT 2005. Further course material as lectures online, lecture notes, proposed and solved exercises will be given through the ‘portale della didattica’ website. 
Assessment and grading criteria
The exam will check your knowledge both theoretical both applied of the material of the course. The exam will consist in a computer assisted test and in a written exam.
The computer test will check your knowledge of the basic aspects of numerical linear algebra and your knowledge of the software Matlab: these aspects have been treated during the lectures and the exercise sessions during the course. The written exam will check your knowledge of the basic aspects linear algebra and geometry: these aspects have been treated during the lectures and the exercise sessions during the course. During your exam you cannot use any electronic device except the PC during in the LAIB. The computer assisted exam is 45 minutes long. The PC will present you with multiple choice questions both practical and theoretical. For the practical questions you will need to use Matlab. There are 8 questions, 6 of them worth 1 point each and 2 of them worth 2 points each. Each question has 5 proposed answers of which only one is the right one. The student must obtain strictly more than 3 points in order to pass; otherwise it will not be possible to take the written exam and the student will fail the exam. The written part will last 60 minutes and it consists of 8 multiple choice questions (worth 2 points each) and one exercise (worth 7 points). The multiple choice questions will check the ablity of doing computation and the theoretical knowledge of the whole nonnumerical part of the program of the course. The exercise will check the ability of solving problems that involves ideas, properties and results explained during the lectures. It is necessary to give the correct answer to at least 3 multiple choice questions, otherwise the student will fail. To pass the exam, the student must attend both parts (computer assisted and written), score the minimal number of required points, and the sum of the two grades (computer assisted + written) must be of at least 18. 
