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. 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 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. 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.

Methods:

- Final oral exam
- Final written exam

Further information:

Final written exam (or 2 x "compitini") and oral exam.

The written exam contains (proof- or computation-based) exercises. The oral exam includes theorem proofs and/or other exercises.

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.

Methods:

- Final oral exam
- Final written exam

Further information:

Final written exam (or 4 x "compitini") and oral exam.

Delivery: face to face

Learning activities:

- attending lectures
- individual study

Attendance: Advised

Teaching methods:

- Lectures
- Task-based learning/problem-based learning/inquiry-based learning

Delivery: face to face

Attendance: Advised

Learning activities:

- attending lectures
- individual study

Teaching methods:

- Lectures
- Task-based learning/problem-based learning/inquiry-based learning

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.

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.

No mandatory reading. Suggested books and resources for individual study include: * The online lecture notes provided by the teachers at http://www.dm.unipi.it/~gaiffi/MatDisc2014/Pages/mat-disc.pdf, http://www.dm.unipi.it/~gaiffi/MatDisc2014/Pages/mainalgebralineare.pdf. * the notes written on the tablet pc during the lectures, available on the instructors' websites. * L. Childs, ALGEBRA, un'introduzione concreta, ETS Editrice. Books covering the topics of the second part of the course (linear algebra): * 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). * prof. M. Gobbino's video lectures on http://users.dma.unipi.it/gobbino/Home_Page/AD_AL_14.html (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).

No mandatory reading. Suggested books and resources for individual study include: * The online lecture notes provided by the teachers at http://www.dm.unipi.it/~gaiffi/MatDisc2014/Pages/mat-disc.pdf, http://www.dm.unipi.it/~gaiffi/MatDisc2014/Pages/mainalgebralineare.pdf. * the notes written on the tablet pc during the lectures, available on the instructors' websites. * L. Childs, ALGEBRA, un'introduzione concreta, ETS Editrice. Books covering the topics of the second part of the course (linear algebra): * 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). * prof. M. Gobbino's video lectures on http://users.dma.unipi.it/gobbino/Home_Page/AD_AL_14.html (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).