Students will become familiar with basic operations involving vector spaces and matrices. They will develop the skills and confidence needed to approach linear systems and eigenvalue problems by: (1) recognizing and expressing linear systems in matrix form; (2) understanding key concepts such as the existence and uniqueness of solutions; (3) gaining the ability to solve such systems using computational tools. Students will also develop an awareness of important computational aspects, such as the efficiency and stability of numerical methods.
Course Prerequisites
No specific prerequisites are required for the course, apart from basic high school-level mathematical knowledge.
Teaching Methods
The major part of the classroom lessons will be held at the blackboard. These lessons will combine theoretical sessions, where new concepts and notions will be introduced, with practical sessions, where exercises will be developed by the lecturer. In addition, a minor part of the lessons will be conducted using a laptop and projector, in order to directly show some computational examples using MATLAB.
Assessment Methods
The exam will consist of a written test. The final grade will follow the standard 0–30 scale, with 18 as the minimum passing mark. During the written examination, no support materials (such as textbooks, notes, or advanced calculators) are allowed. The exam will include both "theoretical questions", aimed at assessing the student’s understanding of the course's theoretical aspects (e.g., stating and explaining a definition or theorem), and "practical exercises", similar to those developed during the practical classroom sessions. The approximate weight of the theoretical part will be 40%, and the practical part 60%.
Texts
The main textbook for the course is: "Linear Algebra and Its Applications", J.C. Lay, R.S. Lay, J.J. McDonald, Pearson. A recommended supplementary text for the computational aspects is: "Numerical Linear Algebra", W. Layton, M.M. Sussman, World Scientific Publishing.
Contents
Vector spaces: linear independence, bases and dimensions, norms and scalar products. Matrices: operations, determinants, rank, nonsingular matrices. Symmetric, positive definite, and orthogonal matrices. Linear maps between vector spaces: kernel, image, rank-nullity theorem, operator norms. Linear systems: direct methods and LU factorization, computational cost. Stability and condition number of linear systems. Least squares problems, QR factorization, computational aspects. Eigenvalues and eigenvectors.
