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This Book is special written according to syllabus of M.Sc Math students. Algebra forms the basis for much of modern mathematics-theoretical, applied, and computational. The purpose of this book is to provide a broad and solid foundation for the study of advanced mathematics. A secondary aim is to introduce the reader to many of the interesting applications of linear algebra..We all are aware that algebra is the part of mathematics dealing with manipulation of expressions and solutions of equations. Since these operations are needed in all branches of mathematics, algebraic skill is a fundamental need for doing mathematics and therefore for working in any discipline which substantially requires mathematics. It is the foundation and the core of higher mathematics. A strong foundation in Algebra will help individuals become better mathematicians, analyzers, and thinkers.