Advanced modern algebra
著者
書誌事項
Advanced modern algebra
Prentice Hall, c2002
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注記
Includes index
内容説明・目次
- 巻冊次
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ISBN 9780130856456
内容説明
For a second applied course in Linear Algebra or a rigorous first course for students of math, physics, engineering, and other sciences.
Built upon the principles of diagonalization and superposition, this text contains many important physical applications-such as population growth, normal modes of oscillations, waves, Markov chains, stability analysis, signal processing, and electrostatics-to show students the incredible power of linear algebra in the real world. The underlying ideas of breaking a vector into modes, and of decoupling a complicated system by suitable choice of linear coordinates, are emphasized throughout the book. It impresses upon students the importance of these principles, while giving them enough tools to use them effectively in a variety of settings.
目次
- 1. The Decoupling Principle. 2. Vector Spaces and Bases. Vector Spaces. Linear Independence, Basis and Dimension. Properties and Uses of a Basis. Change of Basis. Building New Vector Spaces from Old Ones. 3. Linear Transformations and Operators. Definitions and Examples. The Matrix of a Linear Transformation. The Effect of a Change of Basis. Infinite Dimensional Vector Spaces. Kernels, Ranges, and Quotient Maps. 4. An Introduction to Eigenvalues. Definitions and Examples. Bases of Eigenvectos. Eigenvalues and the Characteristic Polynomial. The Need for Complex Eigenvalues. When is an Operator Diagonalizable? Traces, Determinants, and Tricks of the Trade. Simultaneous Diagonalization of Two Operators. Exponentials of Complex Numbers and Matrices. Power Vectors and Jordan Canonical Form. 5. Some Crucial Applications. Discrete-Time Evolution: x(n)=Ax(n-1). First-Order Continuous-Time Evolution: dx/dt=Ax. Second-Order Continuous-Time Evolution: d2x/dt2=Ax. Reducing Second-Order Problems to First-Order. Long-Time Behavior and Stability. Markov Chains and Probability Matrices. Linear Analysis near Fixed Points of Nonlinear Problems. 6. Inner Products. Real Inner Products: Definitions and Examples. Complex Inner Products. Bras, Kets, and Duality. Expansion in Orthonormal Bases: Finding Coefficients. Projections and the Gram-Schmidt Process. Orthogonal Complements and Projections onto Subspaces. Least Squares Solutions. The Spaces l2 and L2(0,1). Fourier Series on an Interval. 7. Adjoints, Hermitian Operators, and Unitary Operators. Adjoints and Transposes. Hermitian Operators. Quadratic Forms and Real Symmetric Matrices. Rotations, Orthogonal Operators, and Unitary Operators. How the Four Classes are Related. 8. The Wave Equation. Waves on the Line. Waves on the Half Line
- Dirichlet and Neumann Boundary Conditions. The Vibrating String. Standing Waves and Fourier Series. Periodic Boundary Conditions. Equivalence of Traveling Waves and Standing Waves. The Different Types of Fourier Series. 9. Continuous Spectra and the Dirac Delta Function. The Spectrum of a Linear Operator. The Dirac o Function. Distributions. Generalized Eigenfunction Expansions
- The Spectral Theorem. 10. Fourier Transforms. Existence of Fourier Transforms. Basic Properties of Fourier Transforms. Convolutions and Differential Equations. Partial Differential Equations. Bandwidth and Heisenberg's Uncertainty Principle. Fourier Transforms on the Half Line. 11. Green's Functions. Delta Functions and the Superposition Principle. Inverting Operators. The Method of Images. Initial Value Problems. Laplace's Equation on R2. Index.
- 巻冊次
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ISBN 9780130878687
内容説明
For two-semester, beginning graduate-level courses in Algebra.
The new "Bibles of Graduate Algebra." This text's organizing principle is the interplay between groups and rings, where "rings" includes the ideas of modules. It contains basic definitions, complete and clear proofs, and gives attention to the topics of algebraic geometry, Groebner bases, homology, and representations. More than merely a succession of definition-theorem-proofs, this text puts results and ideas in context so that students can appreciate why a certain topic is being studied, and where definitions originate.
目次
Preface.
Etymology.
Special Notation.
1. Things Past.
Some Number Theory. Roots of Unity. Some Set Theory.
2. Groups I.
Introduction. Permutations. Groups. Lagrange's Theorem. Homomorphisms. Quotient Groups. Group Actions.
3. Commutative Rings I.
Introduction. First Properties. Polynomials. Greatest Common Divisors. Homomorphisms. Euclidean Rings. Linear Algebra. Quotient Rings and Finite Fields.
4. Fields.
Insolvability of the Quintic. Fundamental Theorem of Galois Theory.
5. Groups II.
Finite Abelian Groups. The Sylow Theorems. The Jordan-Hoelder Theorem. Projective Unimodular Groups. Presentations. The Neilsen-Schreier Theorem.
6. Commutative Rings II.
Prime Ideals and Maximal Ideals. Unique Factorization Domains. Noetherian Rings. Applications of Zorn's Lemma. Varieties. Groebner Bases.
7. Modules and Categories.
Modules. Categories. Functors. Free Modules, Projectives, and Injectives. Limits.
8. Algebras.
Noncommutative Rings. Chain Conditions. Semisimple Rings. Tensor Products. Characters. Theorems of Burnside and Frobenius.
9. Advanced Linear Algebra.
Modules over PIDs. Rational Canonical Forms. Jordan Canonical Forms. Smith Normal Forms. Bilinear Forms. Graded Algebras. Division Algebras. Exterior Algebra. Determinants. Lie Algebras.
10. Homology.
Introduction. Semidirect Products. General Extensions and Cohomology. Homology Functors. Derviced Functors. Ext and Tor. Cohomology of Groups. Crossed Products. Introduction to Spectral Sequences.
11. Commutative Rings III.
Local and Global. Dedekind Rings. Global Dimension. Regular Local Rings.
Appendix A: The Axiom of Choice and Zorn's Lemma.
Bibliography.
Index.
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