Mathematics II or Mathematics applied to social sciences II.

The objective of this course is to familiarize students with the fundamentals of Linear Algebra. More specifically, by the end of the semester, students are expected to be able to: 1. Solve and analyze systems of linear equations using Gaussian elimination. 2. Properly manipulate matrices and vectors (addition; multiplication, computation of inverses and determinants when possible). 3. Determine whether a set of vectors is linearly independent or not. 4. Determine whether a set of vectors forms a vector subspace and, if so, find one of its bases. 5. Determine whether a mapping is linear and, if so, represent it as a matrix with respect to given bases, possibly changing bases. 6. Determine whether an endomorphism can be diagonalized by similarity and, if possible, perform the diagonalization. 7. Understand and work with the abstract concepts of inner product and norm. 8. Obtain orthonormal bases from non-orthogonal ones using the Gram-Schmidt process. 9. Formulate regression problems as least squares problems and solve them using orthogonal projections. 10. Compute the singular value decomposition and the Moore-Penrose inverse of a matrix. 11. Formulate regression problems as least squares problems and solve them using singular value decomposition. 12. Additionally, the course aims to develop the following competencies: 1. Capacity for analysis and synthesis. 2. Modeling and problem-solving skills. 3. Ability to express mathematical reasoning clearly, both orally and in writing.

1. Systems of Linear Equations 1.1. Concept of systems of linear equations 1.2. Gaussian Elimination      1.2.1. Matrix notation      1.2.2. Row reduction and echelon form 1.3. Homogeneous linear systems 1.4. Applications 2. Matrices and Determinants 2.1. Matrices      2.1.1. Matrix operations      2.1.2. Inverse of a matrix      2.1.3. Block matrices      2.1.4. LU factorization 2.2. Determinants      2.2.1. Properties      2.2.2. Cramer¿s Rule 3. Real Vector Spaces 3.1. Vector spaces and subspaces 3.2. Null space and column space      3.2.1. Linear mappings 3.3. Sets of linearly independent vectors. Bases 3.4. Dimension and rank 3.5. Change of basis 4. Eigenvalues and Eigenvectors. Diagonalization 4.1. Eigenvalues and eigenvectors 4.2. Diagonalization 5. Inner Product and Orthogonality. Least Squares Problems 5.1. Inner product, length, orthogonality 5.2. Orthogonal projections 5.3. Gram-Schmidt process 5.4. Least squares problem 6. Singular Values and Vectors. Pseudoinverse 6.1. Symmetric matrices 6.2. Singular value decomposition 6.3. Pseudoinverse or Moore-Penrose inverse 6.4. Applications to least squares problems

The theoretical content of the course will be delivered primarily through lectures. During the time allocated to problem-solving, selected problems from a collection provided to students gradually throughout the semester will be worked on. Student progress in the course will be monitored through assessment tests. These tests will be written exams consisting of several questions related to specific topics previously indicated by the instructor. The tests will take place during regular class hours.