Linear Algebra

Linear algebra is a fundamental mathematical tool with applications in numerous fields of human knowledge: from the natural and behavioural sciences to economics, engineering and computer science, and of course, pure and applied mathematics.
The purpose of this course is to rigorously develop the fundamental concepts of linear algebra, while illustrating its practical usefulness through a representative selection of applications. The aim is not only theoretical understanding, but also the ability to apply these ideas in diverse contexts.
The specific objectives of the course include:
–Becoming familiar with the most basic algebraic structures: vector spaces and linear applications.
–Gaining fluency in the use of vectors, bases, coordinates, basis changes and quotient spaces.
–Master matrix calculus and its relationship to linear applications: operations with matrices, inverse matrices, elementary matrices, rank and solution of systems of linear equations by the Gauss-Jordan method.
–Study determinants: definition, properties and theoretical connection with linear independence, systems of linear equations, rank and invertibility of matrices.
–Study the concepts of eigenvalue and eigenvector, as well as matrix diagonalisation, its conditions of existence and applications.
–Understand the Jordan canonical form of an endomorphism: its existence, calculation and usefulness in the structural study of linear applications.

1. VECTOR SPACES (13 expositive sessions)

Vector spaces. Subspaces. Generators. Elementary operations and sets of generators. Sum and direct sum of subspaces. Linear applications. Quotient space. Linear independence. Bases. Dimension. Supplementary subspaces. Coordinates of a vector. Change of basis.

2. LINEAR APPLICATIONS AND MATRICES (12 expositive sessions)

Matrices. Linear applications and matrices. Basis change matrix. Equivalent matrices. Dual space. Bidual space. Equations of a subspace. Dual homomorphism and transposed matrix. Rank of a matrix. Elementary matrices. Scaled matrices. Reduced scaled form of a matrix. Calculation of the inverse matrix.

3. SYSTEMS OF LINEAR EQUATIONS (3 expositive sessions)

Systems of linear equations. Matrix interpretation. Rouché-Frobenius theorem. Homogeneous systems. Gauss-Jordan method. Discussion of a stepped system.

4. DETERMINANTS (4 expositive sessions)

Multilinear applications. Determinant of a matrix. Properties. Determinants and bases. Existence and uniqueness of the determinant. Determinants and invertible matrices. Determinants and rank of a matrix. Determinants and systems of linear equations. Cramer's rule.

5. CLASSIFICATION OF ENDOMORPHISMS (10 expositive sessions

Eigenvalues and eigenvectors. Characteristic polynomial. Diagonalisation. Minimal polynomial. Invariant subspaces. Cayley-Hamilton theorem. Jordan canonical form.

BASIC:

Axler, S., Linear algebra done right.
Springer, 1995.

Castellet, M.; Llerena, I., Álgebra lineal y geometría.
Ed. Reverté, Barcelona, 1991.

Cohn, P. M. Algebra, Vol. 1(2ª Ed.). Wiley and Sons, Chichester, 1982.

Hernandez, E., Álgebra y geometría.
Ed. Addison Wesley, Madrid, 1994.

COMPLEMENTARY:

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

De Burgos, J., Álgebra lineal y geometría cartesiana.
Ed. MacGraw-Hill, Madrid, 1999.

Godement, R., Álgebra.
Ed. Tecnos, Madrid, 1967.

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

Contribute to achieving the knowledge, skills and competencies set out in the USC Bachelor's Degree in Mathematics: Con01, Con02, Con03, Con04, Con05, Comp01, Comp02, Comp03, Comp04, H/D01, H/D02, H/D03, H/D04, H/D06, H/D07, H/D08, H/D09.

Understand the basic concepts of linear algebra.

Master the algorithms for reducing matrices to staircase form and know how to apply them to rank calculation, basis calculation, solving systems of linear equations, etc.

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

Handle determinants, recognising their great importance at a theoretical level and their limited efficiency in computational practice compared to the Gauss-Jordan method.

Recognise whether a matrix is diagonalizable. Know how to calculate the canonical Jordan form of an endomorphism and apply it to the classification of endomorphisms

The expositive sessions consist of the professor presenting the main results of the subject. Examples will be discussed to facilitate understanding of the content.

For each topic, there will be a set of exercises to be completed in the laboratory sessions, in which students are expected to participate in solving the problems proposed in the exercises and to raise any questions they may have about the theoretical and practical aspects of the subject.

In addition to face-to-face communication, communication with students can also take place via email and the virtual classroom.

The assessment criteria will be continuous assessment combined with a final exam. This final exam will be held on the date set by the Faculty of Mathematics for this purpose.

Continuous assessment will consist of a test that may not be the same for the different groups, but will be coordinated and similar.

The final exam will be the same for both groups.

Both the continuous assessment tests and the final exam will follow the following evaluation system with the indicated weighting
- Written resolution of theoretical questions and written resolution of practical questions: 35%-65%.
- Statement of theorems, development of proofs and resolution of problems/exercises: 35%-65%.

Calculation of the final grade:
The final test, which is compulsory, will be taken in person. The grade for both the first and second attempts will be max{F; 0.25xC + 0.75xF}, where C denotes the continuous assessment grade and F the final test grade.

A student who does not take the final test on either the first or second attempt will be considered as Not Presented.

In cases of fraudulent completion of exercises or tests, the provisions of the Regulations on the assessment of student academic performance and review of qualifications will apply.

Classroom work:

Lectures: 42 hours
Laboratory classes: 42 hours
Tutorials in very small groups: 2 hours
Total: 86 hours

Individual student work: 139 hours


Total hours of work: 225 hours

Study daily with the help of bibliographic material. Read the theoretical part carefully until you have assimilated it and try to answer the questions, exercises or problems presented in the bulletins.