Elements of Abstract and Linear Algebra

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Elements of Abstract and Linear Algebra

Author: E. H. Connell | Size: 0.64 MB | Format: PDF | Quality: Original preprint | Year: 2004 | pages: 146

It covers abstract algebra in general, but the focus is on linear algebra. Here is a brief outline of the six chapters.

Background: Equivalence relations, injective, surjective, and bijective functions, product of sets, unique prime factorization in the integers (18 pages).
Groups: Normal subgroups, quotient groups, homomorphisms, permutations and symmetric groups, product of groups (18 pages).
Rings: Ideals and quotient rings, Zn, homomorphisms, polynomial rings, product of rings, Chinese remainder theorem, characteristic, Boolean rings (16 pages).
Matrices and Matrix Rings: Invertible matrices, elementary operations and elementary matrices, determinant, similarity, trace, characteristic polynomial (14 pages).
Linear Algebra: Modules, homomorphisms, quotient modules, products and coproducts, summands, independence, generating sets, free basis, uniqueness of dimension, change of basis, rank of a matrix, geometric interpretation of determinant, nilpotent homomorphisms, eigenvalues, characteristic roots, inner product spaces, orthogonal groups, diagonalization of symmetric matrices by orthogonal matrices (40 pages).
Appendix: Chinese remainder theorem, prime and maximal ideals, UFDs, splitting short exact sequences, finitely generated modules over Euclidean domains, Jordan form, determinants and multilinear forms, dual spaces (28 pages).

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