Linear Algebra and Analytic Geometry


The student is supposed acquire basic knowledge on Linear Algebra (vide Program) and that, in learning process, logical reasoning and critical mind are developed. 

General characterization





Responsible teacher

António José Mesquita da Cunha Machado Malheiro


Weekly - 5

Total - 70

Teaching language



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


ISABEL CABRAL, CECÍLIA PERDIGÃO, CARLOS SAIAGO, Álgebra Linear, Escolar Editora, 2018 (6th Edition).

T. S. Blyth e E. F. Robertson, Essential student algebra. Volume two: Matrices and Vector Spaces, Chapman and Hall, 1986.

T. S. Blyth e E. F. Robertson, Basic Linear Algebra (Springer undergraduate mathematics series), Springer, 1998.

S. J. Leon, Linear Algebra with Applications, 6th Edition, Prentice Hall, 2002.

J. V. Carvalho, Álgebra Linear e Geometria Analítica, texto de curso ministrado na Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa, Departamento de Matemática da FCT/UNL, 2000.

E. GIRALDES, V. H. FERNANDES e M. P. M. SMITH, Álgebra Linear e Geometria Analítica, McGraw-Hill de Portugal, 1995.

Teaching method

Theoretical classes and pratical classes.

Evaluation method





To obtain approval for the Curricular Unit (UC) of ALGA, the student must have frequency. In this semester, frequency is obtained through weekly training sheets, during the class period, made available through the Moodle platform. Each Formative Form will be available in Moodle for one week. Failure to complete more than two Formative Forms within the established period will result in not obtaining frequency at the UC.


The continuous evaluation consists of carrying out, during the academic period, two in-person and/or distance tests on the Moodle platform, each one quoted from 0 to 20 values ​​(rounded to one decimal place).

All students who, at the time of the test, are enrolled in the ALGA Curricular Unit can take any test.


Let T1 and T2 be the classifications obtained in the 1st and 2nd tests, respectively, rounded to decimals.

The student''s final grade, CF, is obtained by rounding up to the units of (0.4 × T1 + 0.6 × T2), except if the CF is greater than 17, in which case the student may choose to keep the grade at the end of 17 or take a complementary test for grade defense. Approval to the UC occurs whenever CF ≥ 10. If CF<10, the student fails.


All students enrolled in the UC who obtained frequency can take the exam - see point 1.

In this case, the exam grade (NE) replaces the test''s grade to obtain the final grade, CF. The final grade, CF, of the student is obtained by rounding to the units of NE, except if the CF is greater than 17, in which case the student can choose between getting a final grade of 17 or taking a complementary test to defend the grade. Approval to the UC occurs whenever CF ≥ 10. If CF<10, the student fails.


All students wishing to present themselves with a grade improvement must comply with the legal registration formalities for this purpose (information at the Academic Services). The classification of the improvement exam is obtained as indicated in 4. If this result is higher than the one previously obtained in the discipline, it will be taken as the final grade. Otherwise, there is no improvement in grade.


In order for a student to take any of the tests (tests or exams), he/she must register with CLIP at the place and dates mentioned for this purpose.

Tests and examinations will preferably be in person, and in exceptional situations, they may be carried out at a distance if this is possible. In distance tests, video surveillance means using proctoring tools will be used.

On the day of the exam, the student will have to:

  • Identify themselves with Identity Card or Citizen Card;

  • Submit a “Test Notebook” (with the header not filled in) - for face-to-face tests only.


It is up to the person responsible for the UC to resolve situations that are not provided for in these rules, and may resort to consulting other FCT NOVA bodies for aspects where this may be justified.

The regulation was elaborated in accordance with the norms established in the FCT NOVA evaluation regulation available here.

Subject matter



1 – MatricesDefinitions and basic results. Row-echelon form. Matrices and elementary row/column operations. Characterization of invertible matrices and determination of the inverse.

2 – Systems of Linear Equations: Equivalent systems. Matricial representation of a system of linear equations. Resolution and discution of systems.


3 – Determinants: Definition and properties. Determinant of the product. Classical adjoint (adjugate) of a matrix. Computation of the inverse from the adjugate.

4 – Vector Spaces: Definition and properties. Subspaces. Intersection and sum of subspaces and the relation of their dimensions. Linear combinations and subspace generated by a system of vectors. Principal results about linear dependence/independence of a system of vectors. Bases. Extension to a basis of a linearly independent system of vectors.

5 – Linear Transformations: Properties. Dimension theorem and other fundamental results. Matrix of a linear transformation and of composition of transformations. Matrices and changing of bases.

6 – Eigenvalues and Eigenvectors: Eigenvalues and eigenvectors of a matrix/linear operator. Eigenspaces. Algebraic and geometric multiplicity. Diagonalisable matrices/linear operators.

- Inner, Vector and Mixed Products: Definitions and properties in R3.

8 – Analytic Geometry: Cartesian representations of the straight line and the plane. Metric and no metric problems.