Vector Spaces and Matrix Calculus

Linear Algebra is an essential part of the mathematical toolkit required in the modern study of many areas of behavioral, natural, physical and social sciences, in engineering, in business, in computer science, and of course in pure and applied mathematics.
The purposes of this course is to develop the basic concepts of linear algebra and to illustrate their usability by means of a variety of selected applications. More precisely, one can say that the aims are:
i) To provide a first contact with algebraic structures: vector spaces and linear maps as a generalization of vectors in R^3 and matrices, respectively. To learn how to operate with vectors, basis, subspaces and linear maps.
ii)To get acquaintance with the use of matrices in different branches of knowledge.
iii) To understand the need for reducing matrices to predetermined forms and to practice the algorithms.

1.- Vector spaces. (5 theoretical hours)
Definition of vector space: Examples. Subspaces. Quotient spaces. Intersection and sum of subspaces. Systems of generators .

2.- Linear independence and dimension. (6 theoretical hours)
Linear dependence and independence. Bases and dimension. Equations for a subspace. Coordinates. Supplementary subspaces.

3.- Applications between vector spaces. (9 theoretical hours)
Definition of linear map: properties and examples. Subspaces associated to a linear map. The vector space of linear maps. Matrix of a linear map. Change of basis and linear maps.

4.- Matrices. (5 theoretical hours)
Operations with matrices and properties. Non singular matrices. Elementary matrices. Equivalent matrices. Rank of a matrix.

5.- Systems of linear equations. (3 theoretical hours)
Systems of linear equations. Gaussian elimination. The Rouché-Frobenius Theorem.

Basic.

1.-Cohn, P. M. Algebra, Vol. 1(2ª Ed.). Wiley and Sons, Chichester, 1982.
2.-Jeronimo, G., Sabia, J., Tesauri, S. Álgebra lineal. http://mate.dm.uba.ar/~jeronimo/algebra_lineal/AlgebraLineal.pdf.
3.-López Camino, Rafael. Apuntes Geometría I. Curso 2003-2004. Universidad de Granada.

https://www.ugr.es/~rcamino/docencia/geo1-03/g1tema1.pdf
https://www.ugr.es/~rcamino/docencia/geo1-03/g1tema2.pdf
https://www.ugr.es/~rcamino/docencia/geo1-03/g1tema3.pdf


Complementary

1.-Bolos, J.; Cayetano, J.; Requejo, B. Álgebra lineal y Geometría. UNEX, 2007.

2.-Merino, L.; Santos, E. Álgebra lineal con métodos elementales. Thomson, 2006.

Contribute to achieving the basic, general and transversal competences included in the report of the Degree in Mathematics of the USC: CB1, CB2, CB3, CB4, CB5, CG1, CG2, CG3, CG4, CG5, CT1, CT2, CT3, CT5.

Know the basic concepts of Linear Algebra.

Know the algorithms to reduce matrices to row-echelon forms and know how to apply them to the calculation of the range, calculation of base, resolution of systems, etc.

Understand the close relationship between matrices, linear applications and systems of linear equations and be able to use them in different contexts.

We will follow the general methodology for all the subjects of the degree that appears in the Memory of the Degree in Mathematics.

Continuous assessment combined with a final test are planned as evaluation criterion. The final test will be held on the date set by the Faculty of Mathematics for that purpose.

Continuous assessment will consist of one test.

Calculation of the final qualification:

The final test, which will be compulsory, will be face-to-face. The qualification of both the first and the second chance will be the max {F; 0.25xC + 0.75xF} where C is the grade of the continuous assessment and F the grade of the final test.

The assessment system will be the same for the two groups of the subject.

For cases of fraudulent performance of exercises or tests, the provisions of the Regulations for the evaluation of students' academic performance and the review of qualifications will apply.

It will be considered as Not Presented that student that does not show to the final test, both in the first and in the second chance.

Expository classes:: 28
Seminar clases: 14
Laboratory clases: 14.
Tutorials in very small groups: 2
Personal work (non presential) of the student: 92
Total: 150

Study daily with the help of bibliographic material.
Carefully read the theoretical part until it is assimilated and try to answer the questions, exercises or problems presented in the bulletins.