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| MATH0201-2 | Algebra I
- part a) Introduction to Algebra
- part b) Algebra
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| Duration : | part a) Introduction to Algebra : 20h Th
part b) Algebra : 50h Th, 50h Pr
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| Number of credits : |
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| Lecturer : | part a) Introduction to Algebra : Michel Rigo
part b) Algebra : Michel Rigo
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Language(s) of instruction :
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| French language |
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Course contents :
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| As an introduction and toillustrate the course, we present the complex numbers and their main properties. We take this opportunity to recall the leading concepts of mathematical reasoning and mathematical proofs.
 | | part a) Introduction to Algebra |
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| As an introduction and to illustrate the course, we present the complex numbers and their main properties. We take this opportunity to recall the leading concepts of mathematical reasoning and mathematical proofs. |
| | part b) Algebra |
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| The main part of the course is dedicated to the study of finite dimensional linear algebra. We start by introducing the algebraic structures of group, ring and (skew) field. These concepts are illustrated by presenting the ring of integers modulo p. In particular, we can later on consider vector spaces over an arbitrary field. About linear algebra, we start with matrix computations and the theory of determinants and rank of a matrix. Next we study systems of linear equations. We carefully present and discuss the compatibility and the structure of the solutions of such systems. A large part of the course is devoted to vector spaces (linearly independent vectors, base, dimension, sub-space, direct sum). We present linear maps (kernel, range, theorem of dimension,...), eigenvectors and eigenvalues of an endomorphism, (systems of) projectors, dual of a vector space, a fine study of diagonalization, including the nilpotent endomorphisms and Jordan normal form (in the complex case only). We also consider normal, hermitian and unitary matrices and their applications. Finally, polynomials and rational functions are also studied: Gauss' lemma, the fundamental theorem of algebra, Viète's formulas, the ring of polynomials over an arbitrary field and the corresponding ideals (in particular, the notion of principal ideal domain), Descartes' rule. |
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Learning outcomes of the course :
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| | part b) Algebra |
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| At the end of this course, the student should have mastered the rigor of mathematical reasoning and a strong ability to grasp abstract structures and concepts arising in linear algebra. He/she will be able to give arguments about his/her assertions. The student will have at his/her disposal a set of deeply understood theoretical results for which he/she will be able to give a proof. He/she will be able to arrange several results from the course to solve an exercise. The student will easily manipulate and work with classical matrix computations, study the compatibility of a system, give a base of a vector space, make use of matrix representations of linear maps, diagonalization (in particular for normal, hermitian and unitary matrices) and Jordan normal form. Moreover, he/she will easily work with polynomial and rational functions (for instance, finding GCD, asymptotic behavior, decomposing into simple functions,...). In particular, the student will be able to adapt the learned techniques to other contexts appearing in mathematics: geometrical loci, extrema of a function of several variables, applying the theory of diagonalization to solve systems of differential equations, Markov chains, in combinatorics (for instance, give estimate on the number of paths of length n in a graph) or in statistics (like in principal components analysis), computing n-th power of a matrix, ... |
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Prerequisites and co-requisites/ Recommended optional programme components :
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| | part b) Algebra |
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| Perfect knowledge from secondary school is expected. Being trained to abstraction and mathematical reasoning is an advantage. The beginning of this course is dedicated to train these capabilities. |
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Planned learning activities and teaching methods :
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| | part b) Algebra |
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| The practical sessions are mainly dedicated to solve exercises corresponding to the theory considered during the lecture sessions. These sessions are also useful to obtain extra informations or enlightenments on the concepts presented during the lecture sessions.
During those sessions, the class is divided into smaller groups. The schedule will be communicated on the first day of the academic year.
Moreover, the preparation of lists of exercices for the next practical session will be systematically asked . |
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Mode of delivery (face-to-face ; distance-learning) :
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| | part b) Algebra |
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| The theoretical lectures are given three hours a week. The schedule will be communicated on the first day of the academic year. For the practical sessions, a detailed schedule will be given later. |
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Recommended or required readings :
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Assessment methods and criteria :
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| | part b) Algebra |
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| Students are tested all along the year. These tests are aimed to encourage regular work and study and to help students to auto-evaluate themselves. High marks to those tests will be taken into account for the final examination. Bad results to those tests are not taken into account but constitute a serious reminder.
A recapitulative interrogation (written examination) is organized during January. A student succeeding in this test will be exempted of the corresponding subjects for the final examination (in May-June).
The final examination consists of two parts: a written one and an oral one. The written part is devoted to the resolution of problems and exercises. The oral part is devoted to the theory (mainly proofs of theorems) but also includes direct applications of the theory. The January interrogation counts for one third of the final mark. Students who have obtained a mark less than 10 in January will again be asked questions on these particular topics during the May-June session. |
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Organizational remarks :
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| | part b) Algebra |
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| Some useful informations are given on http://www.discmath.ulg.ac.be/ |
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Contacts :
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| | part b) Algebra |
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| M. Rigo
Institut de Mathématique (B37) -
Grande Traverse 12 -
Sart Tilman, 4000 Liège
Tél. : (04) 366.94.87 -
E-mail : M.Rigo@ulg.ac.be |
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| Items online : |
part b) Algebra
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