Linear Algebra III

Beschreibung

These volumes introduce and develop the theory of linear structures for students of mathematics and its applications. Today, linear algebra serves as an essential tool and unifying language across nearly all areas of mathematics. Given its importance in the natural sciences, engineering, and economics, linear algebra is presented as a valuable and widely applicable subject in its own right.

Having developed the theory of linear structures with its geometric and algebraic applications in the first two volumes, this third volume is devotedto the connections to analysis, numerical methods and to applications. The volume begins with an introduction to normed vector spaces and algebras, providing the foundation for the study of convergence and approximation that underlies the subsequent material. 

The treatment of numerical linear algebra focuses on the most important algorithms for systems of linear equations, least-squares problems, and eigenvalue problems.Building on the discussion of Gaussian elimination in Volume 1, this volume introduces iterative methods, including Krylov subspace methods, as well as classical algorithms for eigenvalue computation such as the power, Jacobi, and QR iterations.

Selected applications from mathematics, physics, engineering, and economics are presented. Topics include continuum mechanics, graph theory, dynamical systems, input-output analysis, and Markov chains. A recurring theme is the relationship between discrete and continuous models: starting from discrete-time and discrete-space systems, the discussion progresses naturally to differential equations and, ultimately, to partial differential equations in one and several spatial dimensions. 

In addition to the standard core material, the volume includes topics of interest to students from a variety of disciplines. Students of mathematics education will find an introduction to the basics of iterative methods. Readers inclined toward algebra will encounter connections with graph theory, while students of analysis or physics will benefit from the treatment of normed spaces, dynamical systems, and stochastic matrices. For those specializing in numerical mathematics or optimization, the material on approximation, numerical algorithms, and dynamical systems provides essential preparation for further study. Students of economics and related disciplines will find the discussion of input-output analysis, stochastic processes, and discrete dynamical models especially relevant. 

Particular emphasis is placed on connecting theory and algorithms and on relating both to applications in the sciences. To support this goal, mathematical modelling plays a central role. Ongoing examples drawn from mechanics, electrical networks, and economics are developed progressively alongside the theory. 

These volumes introduce and develop the theory of linear structures for students of mathematics and its applications. Today, linear algebra serves as an essential tool and unifying language across nearly all areas of mathematics. Given its importance in the natural sciences, engineering, and economics, linear algebra is presented as a valuable and widely applicable subject in its own right.

Having developed the theory of linear structures with its geometric and algebraic applications in the first two volumes, this third volume is devoted to the connections to analysis, numerical methods and to applications. The volume begins with an introduction to normed vector spaces and algebras, providing the foundation for the study of convergence and approximation that underlies the subsequent material. 

The treatment of numerical linear algebra focuses on the most important algorithms for systems of linear equations, least-squares problems, and eigenvalue problems. Building on the discussion of Gaussian elimination in Volume 1, this volume introduces iterative methods, including Krylov subspace methods, as well as classical algorithms for eigenvalue computation such as the power, Jacobi, and QR iterations.

Selected applications from mathematics, physics, engineering, and economics are presented. Topics include continuum mechanics, graph theory, dynamical systems, input-output analysis, and Markov chains. A recurring theme is the relationship between discrete and continuous models: starting from discrete-time and discrete-space systems, the discussion progresses naturally to differential equations and, ultimately, to partial differential equations in one and several spatial dimensions. 

In addition to the standard core material, the volume includes topics of interest to students from a variety of disciplines. Students of mathematics education will find an introduction to the basics of iterative methods. Readers inclined toward algebra will encounter connections with graph theory, while students of analysis or physics will benefit from the treatment of normed spaces, dynamical systems, and stochastic matrices. For those specializing in numerical mathematics or optimization, the material on approximation, numerical algorithms, and dynamical systems provides essential preparation for further study. Students of economics and related disciplines will find the discussion of input-output analysis, stochastic processes, and discrete dynamical models especially relevant. 

Particular emphasis is placed on connecting theory and algorithms and on relating both to applications in the sciences. To support this goal, mathematical modelling plays a central role. Ongoing examples drawn from mechanics, electrical networks, and economics are developed progressively alongside the theory. 

Autor*in

Peter Knabner

Themen in »Linear Algebra III«

linear algebra matrix theory numerical linear algebra normed linear spaces and algebras convergence theory Krylov subspace methods Power- Jacobi- and QR-methods Data analysis Linear algebra and graph theory Dynamical systems Input-output analysis Positive matrices Linear Difference and Differential Equations Stochastic matrices

Stimmen zu »Linear Algebra III«

Details

Link teilen