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The student who successfully completes the course will be familiar with induction, simple counting and linear recurrence problems, modular arithmetic, vector spaces, bases and linear independence, determinants, linear maps. They will be able to solve on their own simple problems in these fields and will have developed an intuitive vision of the algebraic and geometric aspects of these topics. They will be able to provide proofs of the basic theoretical results seen in the lectures.
The written exam contains (proof- or computation-based) exercises. Full detail on their solution is required, not only a final result. The oral exam includes theorem proofs and/or other exercises.
- Final oral exam
- Final written exam
Final written exam (or 2 x "compitini") and oral exam.
Delivery: face to face
- attending lectures
- individual study
- Task-based learning/problem-based learning/inquiry-based learning
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Induction. Sequences and linear recurrences.
Divisibility, prime numbers, greatest common divisor, Bézout identity, congruences and modular arithmetic, inverses modulo n, Chinese Remainder theorem, Fermat's little theorem, Euler's phi function, RSA encryption algorithm.
Binomial coefficients, Newton's binomial formula, elementary combinatorics, inclusion-exclusion principle.
Polynomials. Euclidean division of polynomials. polynomial factorization. Complex roots. Irreducible polynomials over the real numbers.
Linear Systems. Matrices. Gauss elimination. Echelon form. Matrix multiplication. Elementary matrices. Inverse matrix.
Vector spaces, linear dependence, dimension and basis. Subspaces. Sum and intersection of subspaces.
Linear applications and associated matrices. Kernel and image of a linear map. Rank. Determinants. Eigenvalues and eigenvectors. Characteristic polynomial of an endomorphism. Diagonalizability.
Orthogonality. Gram-Schmidt orthogonalization.
No mandatory reading. Suggested books and resources for individual study include:
* The online lecture notes provided by the teacher at http://people.dm.unipi.it/berardu/Didattica/2016-17MDAL/MDAL2016-17.html
* L. Childs, ALGEBRA, un'introduzione concreta, ETS Editrice. For the part "induction and arithmetic".
* M. Abate, /Algebra Lineare/, McGraw-Hill. * S. Abeasis, /Elementi di Algebra Lineare e Geometria/, Zanichelli.
* G. Strang, /Introduction to Linear Algebra/, Wellesley Cambridge, or his video lectures on http://ocw.mit.edu (both in English).
(not all of these books are necessary of course; an abundance of sources is listed for the students' benefit, but one or two of these books/resources will cover all the required material).
Short syllabus: Induction; linear recurrences; binomials; pigeonhole principle; gcd, congruences and modular arithmetic; Euclidean algorithm; Chinese remainder theorem. Finite-dimensional vector spaces; subspaces, linear independence and bases; linear applications and associated matrices; inverse matrices and the solution of linear systems; determinants; eigenvalues and eigenvectors.