It is not expected to have completed any subject since this is a first term/first year subject.

1. Make the students understand what linear systems of equations are, where do they usually come from, how they can be represented in matrix form, and how they can be classified, depending on their solution set. 2. Show the students how to solve linear systems of equations using Gaussian elimination, and its connection with the LU factorization. 3. Introduce vector spaces as natural environments where systems of linear equations make sense: 4. Make the students understand that linear transformations are the natural morphisms within the structure of vector space. 5. Introduce eigenvalues and eigenvectors as means to simplify the matrix representation of linear transformations. 6. Define and explore the Jordan canonical form as the simplest possible representation of any given linear transformation

K05: Know the fundamental results of linear algebra, linear geometry and discrete mathematics and how to apply them in applied contexts. S03: Apply mathematical language and abstract-rigorous reasoning in the enunciation and demonstration of results in various areas of mathematics. S13: Formulate real-world problems by means of mathematical models for their subsequent analysis and resolution. S14: Apply appropriate analytical or numerical techniques to solve mathematical models associated with real-world problems and interpret the results obtained. C07: Establish the definition of a new mathematical object, in terms of others already known for solving problems in different contexts.

1. Linear systems of equations 2. Matrices 3. LU factorization 4. Vector spaces 5. Linear transformations 6. Eigenvalues and eigenvectors 7. The Jordan canonical form

LEARNING ACTIVITIES AND METHODOLOGY THEORETICAL-PRACTICAL CLASSES. [44 hours with 100% classroom instruction, 1.76 ECTS] Knowledge and concepts students must acquire. Students receive course notes and will have basic reference texts to facilitate following the classes and carrying out follow up work. Students partake in exercises to resolve practical problems and participate in evaluation tests, all geared towards assessing whether or not the students have acquired the necessary capabilities. PARTIAL TESTS [2 hours with 100% on-site attendance, 0.08 ECTS] Two short continuous evaluation tests (one hour long) will be taken, around the 7th and the 14th teaching weeks, in order to assess whether the students are acquiring the required skills or not. STUDENT INDIVIDUAL WORK OR GROUP WORK [98 hours with 0 % on-site, 3.92 ECTS] FINAL EXAM. [4 hours with 100% on site, 0.16 ECTS] Global assessment of knowledge, skills and capacities acquired throughout the course. METHODOLOGIES THEORY CLASS. Classroom presentations by the teacher with IT and audiovisual support in which the subject`s main concepts are developed, while providing material and bibliography to complement student learning. PRACTICAL CLASS. Resolution of practical cases and problems, posed by the teacher, and carried out individually or in a group. TUTORING SESSIONS. Individualized attendance (individual tutoring sessions) or in-group (group tutoring sessions) for students with a teacher as tutor.