The Theory of a Single Linear Transformation
This chapter is devoted to the study of a single linear transformation in a vector space over a field. We shall obtain a decomposition of the vector spaces into so-called cyclic subspaces relative to a given linear transformation. By choosing appropriate bases in these spaces we obtain certain canonical matrices for the transformation. These results yield necessary and sufficient conditions for similarity of matrices. Following Krull we shall derive the fundamental decomposition theorems by making use of the theory of finitely generated o-modules, o a principal ideal domain. We shall also prove in this chapter the Hamilton-Cayley Frobenius theorems on the characteristic and minimum polynomials of a matrix. Finally we study the algebra of linear transformations that commute with a given transformation.
KeywordsLinear Transformation Characteristic Polynomial Scalar Multiplication Free Module Finite Order
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