# 26509 - Algebra

## DESCRIPTION & CONTEXTUALISATION OF THE SUBJECT

Linear algebra is the branch of mathematics concerning vector spaces, linear transformations between such spaces,
systems of linear equations and matrices.
Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer
science and social sciences.
In this course we are going to develop some concepts learned in secondary school and other concepts will be assumed to
be known.

## COMPETENCIES/LEARNING RESULTS FOR THE SUBJECT

1.Develop the knowledge in Algebra to solve problemas related to Engineering.
2. Apply algebraic procedures to solve Engineering problemas such as cuantitative analysis, employ mathematical
terminology, make abstraction, construct hypothesis, apply and interpret mathematical results and writing a mathematical
proof.
3. Employ computer sources based on the Algebraic knowledge to solve Engineering problems.
4. Follow the right steps to solve problems applying the algebraic procedures, concepts and results.
5. Have a responsible behaviour, being organized and self-sufficient.

## THEORETICAL/PRACTICAL CONTENT

1 lesson: Vector subspaces.
Subspaces. Linear Independence. Basis and dimension. Coordinates and change of basis. Operations between vector
spaces.
2 lesson: Linear transformations.
The null space and range. Special linear transformations. Construction of a linear transformation.
3 lesson: Matrices.
Operations with Matrices. Change of basis matrix. Matrix representation of a linear transformation.
4 lesson: Determinants.
Determinants properties. Calculus of a determinant. Rank of a matrix.The inverse of a square matrix.
5 lesson: systems of linear equations.
Solution Sets of Linear Systems. Methodologies for solving llinear systems. Applications.
6 lesson: Inner Product spaces.
Norm and Orthogonality. Orthogonal basis.
7 lesson: Diagonalization of square matrices..
Diagonalization of Symmetric Matrices.Orthogonal diagonalization.
8 lesson: Conics and quadric surfaces
Reduced equations.

## METHODS

Computer lessons will take place in the Faculty of Engineering (Gipuzkoa) labs.

## TYPES OF TEACHING

Type of teaching M S GA GL GO
Classroom hours 37,5   15   7,5
Hours of study outside the classroom 56,25   22,5   11,25

Legend: M: Lecture S: Seminario GA: Pract.Class.Work GL: Pract.Lab work GO: Pract.computer wo
GCL: Clinical Practice TA: Workshop TI: Ind. workshop GCA: Field workshop

## ASSESSMENT SYSTEMS

- Continuous assessment system

- Final assessment system

## TOOLS USED & GRADING PERCENTAGES

- Extended written exam 70%
- Practical work (exercises, case studies & problems set) 10%
- Individual work 20%

## ORDINARY EXAM CALL: GUIDELINES & DECLINING TO SIT

The evaluation will be carried out according to the article number 8 of the "Normativa reguladora de la evaluación del
alumnado en las titulaciones oficiales de grado". The evaluation will be continous with tasks and activities carried out
along the term. This evaluation will be completed with a final exam that will be held in the exams period.
The tasks and activities will be the following:
Computer lab:
A computer exam will be done for evaluating the computer lessons carried out along the term.
Occasional evaluation of the learning process:
Suggested exercises will be collected by the professor. Exercises will be individual, and if a student does not give the
exercise to the professor the mark will be a 0.
Writing exam:
The exam will consist of theorical questions and with issues related to computer lessons.

Marks:
%10 computer lessons.
%20 occasional evaluation of the learning process.
%70 writting exam.
According to the article 8.3, if a student wants to be evaluated by the system of final evaluation, regardeless his
participation in the continuous evaluatiaon, he must give to the responsible lecturer a resignation letter during. For that,
the students have a period of 9 weeks starting from the beginning of the term. In this case, the final evaluation will consist
of a writing exam (85%) and a computer exam (15%).
The students that does not take the exam will have a "NO PRESENTADO".

## EXTRAORDINARY EXAM CALL: GUIDELINES & DECLINING TO SIT

Writing exam: The exam will consist of theorical questions and with issues related to computer lessons(100%).

## COMPULSORY MATERIALS

The lecturer will give to the students the necessary material.

## BIBLIOGRAPHY

Basic bibliography

S. Lang (1990). Algebra Lineal. México, Addison-Wesley Iberoamericana.
J.C. Del Valle Sotelo (2012). Álgebra lineal para estudiantes de ingeniería y
ciencias. Mexiko: McGraw-Hill.
D.C. Lay (2001). Álgebra lineal y sus aplicaciones.
G. Nakos, D. Joyner (1999). Álgebra lineal con aplicaciones. Barcelona, Thomson.
J. Rojo, I. Martin (1994). Ejercicios y problemas de álgebra lineal. Madrid.
McGraw-HIll.
C. Alcalde, M. C. Caballer (1994). Algebra Lineala. Azterketetako Problema
Ebatziak. Donostia, Elkar
J. Arvesú, F. Marcellán, J. Sánchez (2005). Problemas resueltos de Algebra
J. Burgos (1999). Álgebra Lineal. Madrid: MCGraw-Hill.
S. I. Grossman (1996). Álgebra Lineal con aplicaciones. Mexiko: McGraw-Hill.
E. Hernández (1994). Álgebra y Geometría. Mexiko: Addison-Wesley/UAM.

In-depth bibliography

T. Garcia, A. Ruiz, M. Sainz (1993). Álgebra. Teoría y ejercicios. Madrid, Thomson.
W. K. Nicholson (2003). Álgebra lineal. Madrid, McGraw-Hill.
B. Kolman (1997). Introductory linear algebra with applications. Prentice-Hall.
I. Kostrikin (1992). Introducción al Álgebra Lineal. Madrid, McGraw-Hill.
J.L. Malaina, A. Gallego, M. L. González, E. Martín (1992). Lecciones básicas de
Álgebra Lineal. Bilbo, EHUko Argitalpen Zerbitzua.
D. Poole (2004). Álgebra Lineal. Una Introducción moderna. Madril, Thomson.
G. Williams (1999). Álgebra Lineal y sus aplicaciones. Mexiko, Wm. C. Brown
Publishers.

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