• Matrix algebra-basic properties. Invertible matrices and their properties. • Determinants and their properties. Computing the inverse of a matrix. • Matrices and systems of linear equations. Rank of a matrix. Solving linear systems-Algorithms Gauss and Cramer. • Vector spaces and subspaces. Special vector subspaces (sum, intersection, orthogonal complement). Linear combinations, linear dependence-independence of vectors. Basis and dimension of a vector space. • Linear transformation – Kernel and Image of a linear transformation. Rank–nullity theorem. Matrix of a linear transformation. Similar matrices. • Inner product on a vector space. Norm of a vector. Orthonormalization of a basis. Gram-Schmidt algorithm. Orthogonal spaces and basic theorems. • Properties of eigenvalues and eingenvectors of a matrix. Theorem Cayley- Hamilton. Minimal polynomial. • Canonical forms. Diagonalizing of a matrix. Criteria diagonalizable matrix (linear transformation). Spectral theorem. Applications. • Quadratic forms-basic criteria for symmetric matrices. Applications to quadratic forms on the problems min-max of a function of several real variables.
Since the topic of the course is the study of fundamental concepts of the linear algebra such as vector spaces and linear transformation, the matrix theory is an essential tool in the study and modeling of many mathematical problems and the concept of the matrix is unfamiliar to the student, the main aim of the course is:
The course aims at the acquisition of knowledge and skills so that they can be applied to other courses of mathematics and computer science.
Mandatory written exams at the end of the semester.