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 original space. This is what is the idea behind “diagonalization”. But we know that all linear transformations are not diagonalizable and then it becomes necessary to understand which transformations are diagonalizable and when. Further if the matrix of the linear transformation is not diagonalizable it might still be reducible into a form that is “block-diagonal” and we need to see what is the simplest block-diagonal form to which a matrix can be reduced. Here I shall list out in a logical sequence the theorems which establish the above things and I shall indicate the basic idea behind them omitting the detailed proofs. All vector-spaces in the following section are finite dimensional. Many of the concepts might not naturally extend for infinite dimensional vectorspaces. 3 elementary operations on vector spaces: