Undergraduate Algebra: A First Course
Designed for second-year mathematics students, this book offers a modern, highly systematic approach to thoroughly familiarize students with the theory of rings, fields, vector spaces, and particularly with the techniques of matrix manipulation. In keeping the needs of the learner paramount, the author provides motivation at each difficult point and integrates a wide range of exercises into each chapter. The method is both strong in its presentation of linear algebra and relevant to computer science.
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Rings fields and complex numbers
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1-element addition and multiplication belong bijective called characteristic polynomial characteristic roots characteristic vectors coefficient column commutative ring complex numbers congruence Corollary corresponding coset deduce defined Definition Let denote determinant elements entries equation equivalence relation Euclidean space example expressed factors field F finite finite-dimensional form a basis hence hermitian instance integer integral domain invertible matrix irreducible polynomials isomorphism linear combination linear mapping linearly independent m x n matrix over F modn modulus monic n x n matrix natural number notation obtain orthogonal orthonormal basis permutations Proof Let Proposition Let quadratic rank reader real numbers ring homomorphism row space row-equivalent row-reduced echelon form satisfies scalar multiplication Show solution span subgroup subring subsets suppose surjective theorem unique unitary space vector space verify write zero