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

António José Mesquita da Cunha Machado Malheiro


Weekly - 6

Total - 72

Teaching language



This course has no prerequisites.


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, 2018 (5ª Edição).

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

Students registered for the 1st time must attend all classes, with possible exception of 3 of them, while the other students must attend 2/3 of the classes.

There are two mid-term tests. These tests can substitute the final exam if the student has a grade CT of at least, 9.5.  CT is calculated as follows:

 CT = 0,5*T1 + 0,5*T2

 where Ti, 1 ≤ i ≤ 2, is the non-rounded grade obtained in test i.

If the student satisfies the conditions above with CT (rounded to units) greater than 16, he may choose between having 16 as final grade or undertake a complimentary assessment.

To be approved in the final exam, the student must have a minimum grade of 9.5 in it.  Again, for grades (rounded to units) greater than 16, the student must undertake a complimentary assessment, otherwise, his final grade will be 16.

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: