Applied Linear Algebra

The Decoupling Principle, Second Edition

Lorenzo Sadun

• Publisher: American Mathematical Society
• Year: 2007
• Print ISBN:
  9781470470043
• eBook ISBN:
  9781470412005
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Linear algebra permeates mathematics, as well as physics and engineering. In this text for junior and senior undergraduates, Sadun treats diagonalization as a central tool in solving complicated problems in these subjects by reducing coupled linear evolution problems to a sequence of simpler decoupled problems. This is the Decoupling Principle.Traditionally, difference equations, Markov chains, coupled oscillators, Fourier series, the wave equation, the Schrödinger equation, and Fourier transforms are treated separately, often in different courses. Here, they are treated as particular instances of the decoupling principle, and their solutions are remarkably similar. By understanding this general principle and the many applications given in the book, students will be able to recognize it and to apply it in many other settings.Sadun includes some topics relating to infinite-dimensional spaces. He does not present a general theory, but enough so as to apply the decoupling principle to the wave equation, leading to Fourier series and the Fourier transform.The second edition contains a series of Explorations. Most are numerical labs in which the reader is asked to use standard computer software to look deeper into the subject. Some explorations are theoretical, for instance, relating linear algebra to quantum mechanics. There is also an appendix reviewing basic matrix operations and another with solutions to a third of the exercises.

Table of contents

• Cover
• Copyright
• Dedication
• Contents
• Preface to second edition
• Preface
• Chapter 1. The Decoupling Principle
• Chapter 2. Vector Spaces and Bases
• Chapter 3. Linear Transformations and Operators
• Chapter 4. An Introduction to Eigenvalues
• Chapter 5. Some Crucial Applications
• Chapter 6. Inner Products
• Chapter 7. Adjoints, Hermitian Operators, and Unitary Operators
• Chapter 8. The Wave Equation
• Chapter 9. Continuous Spectra and the Dirac Delta Function
• Chapter 10. Fourier Transforms
• Chapter 11. Green’s Functions
• Appendix A. Matrix Operations
• Appendix B. Solutions to Selected Exercises
• Index
• Back cover