DECOMPOSITIONS OF COUNTABLE LINEAR TRANSFORMATIONS

A journey through decompositions of linear transformations

For the cause of Representation Theory it is important to understand the elementary ideas that go into the idea of decomposition of a linear transformation into transformations on smaller dimensional vector-spaces. It is desirable that a given linear transformation on an ndimensional vector-space is writable as a ”direct sum” of n linear transformations on one-dimensional subspaces of the origi...

On One-Sided Ideals of Rings of Linear Transformations

6 Countable Primitive Linear Groups

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings, oft...

Solving System of Linear Congruence Equations over some Rings by Decompositions of Modules

In this paper, we deal with solving systems of linear congruences over commutative CF-rings. More precisely, let R be a CF-ring (every finitely generated direct sum of cyclic R-modules has a canonical form) and let I_1,..., I_n be n ideals of R. We introduce congruence matrices theory techniques and exploit its application to solve the above system. Further, we investigate the application of co...

Linear Transformations

The n×m matrix AT obtained by exchanging rows and columns of A is called the transpose of A. A matrix A is said to be symmetric if A = AT . The sum of two matrices of equal size is the matrix of the entry-by-entry sums, and the scalar product of a real number a and an m× n matrix A is the m× n matrix of all the entries of A, each multiplied by a. The difference of two matrices of equal size A a...