Linear algebra is an essential tool in many branches of mathematics and has wide applications. A large part of the subject consists of the study of homomorphisms of (finitely generated) free modules (in particular, linear transformations of finite dimensional vector spaces). There is a crucial relationship between such homomorphisms and matrices (Section 1). The investigation of the connection between two matrices that represent the same homomorphism (relative to different bases) leads to the concepts of equivalence and similarity of matrices (Sections 2 and 4). Certain important invariants of matrices under similarity are considered in Section 5. Determinants of matrices (Section 3) are quite useful at several points in the discussion.
KeywordsLinear Transformation Commutative Ring Division Ring Minimal Polynomial Invariant Factor
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