Elements of Analysis and Algebra I

General characterization

Prerequisites

The student must be familiar with mathematics taught at pre-university level in Portugal (science area).

Bibliography

[1] Cabral, I. & Perdigão, C. & Saiago, C., Álgebra Linear, Escolar Editora, 2008.

[2] Blyth, T. S. & Robertson, E. F. - Basic Linear Algebra, Springer Verlag, 1998.

[3] Anton, H. & Rorres, C. - Elementary Linear Algebra, Applications version, 9th Edition, John Wiley & Sons, Inc., 2005.

[4] Sá, A. & Louro, B. – Análise Matemática, Teoria e Exercícios. (Departamento de Matemática da F.C.T-U.N.L.) 

[5] Sá, A. & Louro, B. – Sucessões e Séries, Teoria e Prática. Escolar Editora, 2008. 

Teaching method

The lectures consist on the theoretical exposition of the syllabus, illustrated with examples.The practical classes consist on the resolution of exercises on all the contents.

Any doubts are clarified during classes, in weekly scheduled sessions, or in extra sessions accorded directly between student and teacher.

There aretwo mid-term tests that can substitute the final exam in case of approval. Otherwise the student must pass the final exam.

Evaluation method

1. FREQUENCY

Any student enrolled in the Curricular Unit (including repeat students who did not obtain attendance in the 2023/2024 academic year) who does not have, in CLIP, the status of student worker and who has been absent for more than 1/3 (one third) of the practical classes taught, he does not obtain frequency and, consequently, he fails the Curricular Unit.

For a student to take any of the tests, they will have to register in CLIP at the location and dates mentioned for that purpose. On the day of the test, the student must bring with them:

a) Identification Document.
b) Test booklet (with the header not filled in).

2. CONTINUOUS EVALUATION

Continuous assessment consists of carrying out, during the semester, 2 theoretical-practical tests, in person, lasting 90th minutes each, in the form of a written test, each of which is rated from 0 to 20 points.
Let T1 and T2 be the classifications obtained in the 1st and 2nd tests, respectively. A student can only pass the subject through continuous assessment if

0.5×T1 + 0.5×T2 ≥ 9.5

In this case, the final classification will be given by this average rounded to the nearest units.

3. EXAM

All students enrolled in the Curricular Unit can take the exam, except for students who have not obtained frequency.

The theoretical-practical exam consists of taking a written test, in person, lasting 3 hours, with a rating of 0 to 20 points.

If the classification is less than or equal to 9.4, the student fails. If the classification is greater than or equal to 9.5, the student is approved with that classification, rounded to the nearest number.

4. GRADE IMPROVEMENT

Any student who wishes to improve their grade must register for this purpose in CLIP (information in the Academic Office). The classification of the improvement exam is obtained as indicated in 3. If this result is higher than the one previously obtained in the subject, it will be taken as the final grade. Otherwise, there will be no grade improvement.

Note: when taking any of the tests (tests or exams), the use of a calculator or any type of consultation is not permitted.

Subject matter

Part I - Linear Algebra

1- Matrices: algebra of matrices, inverse of a matrix.

2- Systems of linear equations: resolution and discussion.

3- Determinants: definition and properties of determinant, inverse matrix calculation from the adjoint matrix, Cramer''''''''s rule.

4- Eigenvalues and eigenvectors​​: linear function, eigenvalues and ​​eigenvectors of a matrix.

Part II - Analysis

1-Limits and continuity: limits and continuity of functions of one variable.

2-Differential calculus: derivative of a function, Taylor''''''''s theorem and applications of Taylor''''''''s formula.

3-Integral Calculus: antiderivatives, Riemann integral.

 

Programs

Programs where the course is taught: