Linear Algebra 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 - 6

Total - 72

Teaching language



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


Main bibliography:

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

2. I. Cabral, C. Perdigão, C. Saiago, Álgebra Linear, Escolar Editora, 6ª Edição, 2021 (ou  5ª Edição 2018) .

3. T. S. Blyth, E. F. Robertson, Basic Linear Algebra, 2nd Edition, Springer Undergraduate Mathematics Series, 2002.

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

5. S. J. Leon, Linear Algebra with Applications, 7th Edition, Prentice Hall, 2006.

6. A. Monteiro, Álgebra Linear e Geometria Analítica, McGraw-Hill de Portugal, 2001.

7. A. P. Santana, J. F. Queiró, Introdução à Álgebra Linear, Gradiva, 2010.

Teaching method

Classes consist on an oral explanation of the theory which is illustrated by examples and the resolution of some exercises.

Results are proven.

Students have access to copies of the theory and proposed exercises. Some of the exercises are solved in class, the remaining are left to the students as part of their learning process.

Any questions will be clarified in class, in weekly scheduled sessions or in special sessions accorded 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. Matrix. Special types of matrices. Basic operations. Row echelon form  and reduced row echelon form.  Elementary row operations and elementary column operations. Rank. Elementary matrix. Invertible matrix, its inverse and algorithm to calculate the inverse.

 2. System of linear equations. Matrix representation of a system. How to discuss and how to solve a system using matrices. Homogeneous system. 

 3. (Real or complex) Vector space. Subspace. Intersection of subspaces. Subspace spanned by a finite sequence of vectors. Linear independence. Bases and dimension. Sum and direct sum of subspaces. Row-space and column-space of a matrix.

4. Linear transformation. Kernel and range. Extension by linearity Theorem. Matrix of a linear transformation (bases fixed) and applications of this notion. Transition matrix. Relationship between matrices of a given linear transformation through transition matrices. 

5. Determinants. Properties and applications.


Programs where the course is taught: