Linear Algebra and Analytic Geometry I


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

General characterization





Responsible teacher

Carlos Manuel Saiago


Weekly - 5

Total - 60

Teaching language



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



[Ant2000] H. Anton, C. Rorres, Elementary Linear Algebra, Applications Version, 11th Edition, John Wiley & Sons, 2000.

[Cab2021] I. Cabral, C. Perdigão, C. Saiago, Álgebra Linear, Escolar Editora, 6ª Edição, 2021 (ou  5ª Edição 2021.

[Car2000] 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.

[Gir1995] E. Giraldes, V.H. Fernandes, M.P. Marques-Smith, Álgebra Linear e Geometria Analítica, McGraw-Hill de Portugal, 1995.

[Lay2016] D. Lay, S. Lay, J. McDonald, Linear algebra and its applications, 5th edition, Pearson, 2016.

[Leo2006] S.J. Leon, Linear Algebra with Applications, 7th Edition, Prentice Hall, 2006.

[Mon2001] A. Monteiro, Álgebra Linear e Geometria Analítica, McGraw-Hill de Portugal, 2001.

[San2010] P. Santana, J.F. Queiró, Introdução à Álgebra Linear, 2010.

Teaching method


There are classes in which theory is lectured and illustrated by examples.

There are also problem solving sessions. Some exercises are left to the students to be solved on their own as part of their learning process. Students can ask questions during the classes, in weekly scheduled sessions or in special sessions accorded directly with the professor.

Evaluation method

There are two 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


1-2. Systems of linear equations and matrices: basic definitions; algebraic, elementary; matrix rank; Gauss and Gauss-Jordan elimination method; matrix inversion.

3. Determinants: definition; Laplace theorem; application to matrix inversion; Cramer’s rule.

4. Vector spaces: definition; vector subspaces; basis and dimension; sum and direct sum of vector spaces; change of basis.

5. Linear transformations: definition; matrix representation; kernel and image; isomorphisms; composition and invertibility; change of basis.

6. Eigenvalues and eigenvectors: definitions; proper subspaces; characteristic polynomial; algebraic and geometric multiplicity; matrix diagonalization; real symmetric matrix diagonalization.

7. Euclidean spaces: inner products; norm, Cauchy-Schwarz inequality, angle; orthogonal projections; orthogonal basis; Gram-Schmidt orthogonalization; orthogonal complement; cross and mixed products of vectors in R^3.