Algebra: ring and field theory. Lecture 15, math 100 c, UCSD

Using the existence of a Smith normal form over a Euclidean domain, we prove a classification of finitely generated modules over a Euclidean domain. As a special case, we obtain a classification of finitely generated and finite abelian groups.

In the second half, we defined a matrix representation of a linear map, and formulated a question: can we find a sparse matrix representation of a linear map? We start with the evaluation map from the ring of polynomials to the ring of linear maps. Similar to the study of algebraic numbers, the minimal polynomial of a linear transformation is defined. We also show that two different matrix representations of a given linear map are conjugate of each other. We define minimal polynomial of a square matrix and show that the minimal polynomial does not change after conjugation. Finally we say how having a linear map T from V to V, we can view V as a F[x]-module. This will be used later.

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