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

Herberto de Jesus da Silva


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 (5th 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






 Continuous assessment consists of conducting, during the semester, 2 online tests in Moodle, each of which is rated from 0 to 20 points.

All students who are enrolled in the Curricular Unit of Linear Algebra and Analytical Geometry can present themselves for any test.

Let T 1 and T 2 be the classifications obtained in the 1st and 2nd tests, respectively. A student can only pass the subject by continuous assessment if

 0,4×T 1  + 0,6×T 2  ≥ 9,5 .

 In this case the final classification will be given by this average rounded to the units, except if this average is greater than or equal to 17.5, in which case the student may choose to stay with the final classification of 17 or take a complementary test to defend the classification.

 2. EXAM

 All students enrolled in the Course can take the exam.

If the grade is less than or equal to 9.4, the student fails. If the classification is higher or equal to 9.5 and lower or equal to 17.4, the student is approved with this classification, rounded to the nearest integer. If the classification is higher or equal to 17.5, the student may choose to stay with the final classification of 17 or take a complementary test to defend his / her grade.


 Any student wishing to present a grade improvement must register for this purpose at CLIP (information at the Academic Office). The classification of the improvement exam is obtained according to what is indicated in 2. If this result is higher than that already obtained in the discipline, it will be taken as a final grade. Otherwise, there is no grade improvement.

Subject matter


1 – Matrices: Definitions 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.

7 - 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.