The student will become familiar with the concepts of: 1- Linear systems. 2- The algebra of matrices and vectors. 3- Vector subspaces in Rn. 4- Complex numbers. The student will acquire the skills to be able: 1- Calculate the solution of a system of linear equations 2- Discuss the existence and uniqueness of solutions of a system of linear equations 3- Operate with vectors and matrices 4- Calculate the inverse of a matrix 5- Calculate bases of vector subspaces 6- Understand and operate with linear transformations 7- Calculate eigenvalues and eigenvectors of a matrix 8- Calculate an orthonormal base from any basis 9- Solve least-squares problems 10- Calculate a unitary diagonalization of a normal matrix

1. Complex numbers · Numbers sets · Necessity of complex numbers · Binomial form of a complex number · Graphical representation · Operations · Complex conjugate, modulus, argument · Polar form of a complex number · Roots of complex numbers · Exponential of a complex number · Solving equations 2. Vector spaces · Definition · Subspace · Linear combinations · Linear independence · Basis of a vector space · Span · Linear map · Linear equations · Linear transformations 3. Matrix algebra · Definition · Connection to linear map · Column space · Row space · Row echelon form · Rank of a matrix · Basis transformations 4. Solving linear systems · Row echelon form · Existence and uniqueness · Null space · Gauss elimination · Matrix inverse · Determinant 5. Eigenvalue problems · Definition of eigenvalue and eigenvector · Properties · Characteristic equation · Diagonalization 6. Euclidean vector space · Inner product · Geometric interpretation · Orthogonal projection · Gram-Schmidt orthogonalization procedure · QR decomposition · Unitary matrices · Normal matrices 7. Least squares problems · Normal equations · Relation to data fitting

The teaching methodology will include - Theory classes, where the knowledge that students must acquire will be presented. A textbook (Linear Algebra and its Applications, by David C. Lay) will be followed to facilitate its development. Students will receive the course syllabus and are expected to prepare classes in advance. - Resolution of exercises by the student that will serve as self-evaluation and to acquire the necessary skills. - Problem classes, in which the problems proposed to the students are developed and discussed. - The teacher may pose problems and work to solve individually or in group. - The teacher will set his schedule of individual tutorials.