The student is expected to know and understand the fundamental concepts of: - Systems of linear equations - Matrix and vector algebra - Vector spaces - Linear maps - Eigenvalues and eigenvectors - Jordan canonical form The student is expected to acquire and develop the ability to: - Discuss the existence and uniqueness of solutions of a system of linear equations - Solve a consistent system of linear equations - Carry out basic operations with vectors and matrices - Compute the LU factorization of a matrix - Determine whether a square matrix is invertible or not, and compute the inverse matrix if it exists - Determine whether a subset of a vector space is a subspace or not - Find bases of a vector subspace, and compute change-of-basis matrices - Compute eigenvalues and eigenvectors of a square matrix - Determine whether a square matrix is diagonalizable or not - Diagonalize a matrix - Compute the Jordan form of a matrix

K1: To know the main techniques of mathematical proof, as well as to understand the importance and necessity of hypotheses in mathematical results. K2: Know the fundamental definitions and results of algebra, geometry and discrete mathematics, including both the statements and their proofs. S1: Learn and adapt mathematical techniques and methods from one branch to another (such as algebra, calculus or probability) and apply them to different scientific or industrial problems. S4: Use logical and abstract reasoning to state, demonstrate and verify the validity of mathematical results, as well as to analyze models and design solution strategies. C3: Use numerical or symbolic calculation, statistical analysis, or optimization software to approximate the solution of mathematical problems arising in a professional context and know how to analyze and predict behaviors in different contexts, implementing efficient solutions to complex problems.

. Matrices and vectors . Systems of linear equations . LU factorization . Vector spaces . Linear maps . Eigenvalues and eigenvectors . Jordan canonical form

The teaching methodology will include: - Theoretical lectures in large groups, where knowledge that students should acquire will be presented. The course weekly schedule will be available to students and they are expected to prepare the classes in advance. - Resolution of exercises by the student, which will serve them as a self-assessment and to acquire the necessary skills - Problem classes, during which problems are discussed and solved - Tutorships