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 - 41

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

There are three tests that can substitute the final exam in case of approval. Otherwise the student must succeed the final exam. More detailed rules are available in the Portuguese version.

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.