A survey of matri̇x groups in lie theory

Matrix groups have an important role in many fields of mathematics and its applications. Matrix groups are discussed especially with their algebraic structures (in the sense of abstract algebra and linear algebra), in this thesis the topological structure of matrix groups is also discussed. It has also been seen in previous studies that matrix groups can be seen as an interesting mathematically intertwined example of algebra and geometry. Lie theory is a subbranch of mathematics consisting of Lie groups and Lie algebras and their applications, and was introduced by the Norwegian mathematician Sophus Lie in the late 19th century. It is basically based on the theory of continuous or differentiable abstract groups. In the thesis, groups of matrices corresponding to linear transformations on n-dimensional Euclidean space are discussed. For doing this, the structures of matrix groups from various sources and their related propositions and theorems are used. On the other hand, Lie theory is mainly based on topology and group structure of properly differentiable manifolds. For this reason, for the sake of completeness, topological foundations related to the related definitions and theorems in the theory of differentiable manifolds are mentioned. A Lie group is associated with its associated structure, the concept of Lie algebra. The relationship between Lie groups and Lie algebras is provided by exponential map. In some studies, when working with Lie groups, it may be more practical and appropriate to work with Lie algebras instead. Any matrix group has a manifold and a Lie group structure with the operation, along with the differentiable structure we build on it. In this thesis, the general linear group, the special linear group and the orthogonal group have been dealt with and their Lie algebras have been examined in detail.