Elementary notions on mathematics, including: - Solution of linear systems (at least 3 equations) - Evaluation of functions (in particular, of polynomial functions) - Vectors in R^2 and R^3. Elementary euclidean geometry in R^2 and R^3 (inner product, euclidean distance, modulus of a vector) - Basic matrix theory.

By the end of this subject, students will be able to have: 1. Knowledge and understanding of the mathematical principles of linear algebra underlying Mechanical Engineering; 2. The ability to apply their knowledge and understanding to identify, formulate and solve mathematical problems of linear algebra using established methods; 3. The ability to select and use appropriate tools and methods to solve mathematical problems using linear algebra; 4. The ability to combine theory and practice to solve mathematical problems of linear algebra.

RA1.1 Knowledge and understanding of the scientific and mathematical principles underlying their branch of engineering. RA2.1 The ability to apply their knowledge and understanding to identify, formulate and solve engineering problems using established methods. RA5.1 The ability to select and use appropriate equipment, tools and methods. RA5.2 The ability to combine theory and practice to solve engineering problems. CB1 Students have demonstrated possession and understanding of knowledge in an area of study that builds on the foundation of general secondary education, and is usually at a level that, while relying on advanced textbooks, also includes some aspects that involve knowledge from the cutting edge of their field of study. CB2 Students are able to apply their knowledge to their work or vocation in a professional manner and possess the competences usually demonstrated through the development and defence of arguments and problem solving within their field of study. CG1 Ability to resolve problems with initiative, creativity decision-making and critical reasoning skills, and to communicate and transmit knowledge, skills and abilities in the Industrial Engineering area. CG11 Capacity to solve mathematic problems arising in engineering. Aptitude for applying knowledge of: linear algebra; geometry; differential geometry; differential and integral calculus; differential equations and partial derivatives: numerical methods; numerical algorithms; statistics and optimization.

Topic 0.- Complex numbers. Topic I.- Systems of linear equations. Introduction to systems of linear equations. Elementary operations per row. Reduction by rows and staggered forms. Vectorial equations. The matrix equation Ax = b. Solution sets of linear systems. Topic II.- Matrix Algebra. Operations with matrices. The inverse of a matrix. Matrices divided by blocks. Determinants. Topic III.- Vector spaces. Vector spaces and subspecies. Linearly independent sets and bases. Coordinate systems and dimension. Linear transformations. Topic IV.- Values and eigenvectors. Introduction to eigenvalues and eigenvectors. Diagonalization of square matrices. Topic V.- Orthogonality and least squares. Scalar product, norm and orthogonality. Orthogonal sets. Orthogonal projections and least squares problems. Topic VI.- Symmetric Matrices.

The teaching methodology is based on the "flipped classroom" approach, and it will consist of: - Theoretical and practical videos, that the student should visualize before attending the classes. - Solving exercises and problems sessions. These sessions should be participative, and the students are expected to take an active part in these sessions. - The weekly planning of the course, together with the necessary material, will be available to the students in advance, allowing the students to prepare the classes beforehand. The methodology will also contain: - Solving exercises by the student, which will serve as self-assessment and to acquire the necessary skills. - Tutorial sessions, individual and voluntary, in which students will have the possibility to consult the teacher their doubts and questions on the subject. The time and place of these sessions will be set by the teacher at the beginning of the course.