Linear Algebra


Linear Algebra is an essential tool in many branches in mathematics and has wide applications. A large part of the subject consists of the study of homomorphism of (finitely generated) free modules (in particular, linear transformations of finite dimensional vector spaces). There is crucial relationship between such homomorphism and matrices. The investigation of the connection between two matrices that represent the same homomorphism (relative to different bases) leads to the concepts of equivalence and and similarity of matrices. Certain important invariants of matrices under similarity are considered. Determinant of matrices are quite useful at several point in the discussion.
Since there is much interest in the applications of linear algebra, a great deal of material of calculational nature included in this chapter. For many readers the inclusion of such material will be well worth the burden of additional length. However, the chapter is so arranged that the reader who whises only to cover the important basic facts of the theory may do so in a relatively short time. He need only omit those result labeled as propositions and observe the comments in the text as to which material is needed in the sequel.
(Taken from Graduate Texts in Mathematics, Abstract Algebra, Thomas W. Hungerford, 1974)

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