Orthogonal Sets and Polar Methods in Linear Algebra: Applications to Matrix Calculations, Systems of Equations, Inequalities, and Linear Programming ... Wiley Series of Texts, Monographs and Tracts)

A unique, applied approach to problem solving in linear algebra Departing from the standard methods of analysis, this unique book presents methodologies and algorithms based on the concept of orthogonality and demonstrates their application to both standard and novel problems in linear algebra. Covering basic theory of linear systems, linear inequalities, and linear programming, it focuses on elegant, computationally simple solutions to real-world physical, economic, and engineering problems. The authors clearly explain the reasons behind the analysis of different structures and concepts and use numerous illustrative examples to correlate the mathematical models to the reality they represent. Readers are given precise guidelines for: \* Checking the equivalence of two systems \* Solving a system in certain selected variables \* Modifying systems of equations \* Solving linear systems of inequalities \* Using the new exterior point method \* Modifying a linear programming problem With few prerequisites, but with plenty of figures and tables, end-of-chapter exercises as well as Java and Mathematica programs available from the authors' Web site, this is an invaluable text/reference for mathematicians, engineers, applied scientists, and graduate students in mathematics.Content: Chapter 1 Basic Concepts (pages 1–22): Chapter 2 Orthogonal Sets (pages 23–40): Chapter 3 Matrix Calculations Using Orthogonal Sets (pages 41–72): Chapter 4 More Applications of Orthogonal Sets (pages 73–84): Chapter 5 Orthogonal Sets and Systems of Linear Equations (pages 85–142): Chapter 6 Polyhedral Convex Cones (pages 143–190): Chapter 7 Polytopes and Polyhedra (pages 191–213): Chapter 8 Cones and Systems of Inequalities (pages 215–250): Chapter 9 An Introduction to Linear Programming (pages 251–274): Chapter 10 The Exterior Point Method (pages 275–341): Chapter 11 Applications (pages 343–391): Linear Spaces And Systems Of Equations -- Basic Concepts -- Orthogonal Sets -- Matrix Calculations Using Orthogonal Sets -- More Applications Of Orthogonal Sets -- Orthogonal Sets And Systems Of Linear Equations -- Cones And Systems Of Inequalities -- Polyhedral Convex Cones -- Polytopes And Polyhedra -- Cones And Systems Of Inequalities -- Linear Programming -- An Introduction To Linear Programming -- The Exterior Point Method -- Applications -- Applications -- Appendices -- References -- Index. Enrique Castillo ... [et Al.]. A Wiley-interscience Publication. Includes Bibliographical References (p. 415-418) And Index. This unique, non-standard approach to solving problems in linear algebra discusses topics in four major parts, covering the basic theory of linear systems, solving linear inequalities, linear programming, and applications. It correlates physical and mathematical realities, clearly explaining the reasons behind the analysis of different structures and concepts. This is a first-year graduate text on linear algebra for applications. Castillo makes a great effort at connecting physical and mathematical realities, showing the reader the reasons that lead to the analysis of structures and concepts In this chapter we introduce some concepts and definitions that are needed for the following chapters, such as linear spaces, subspaces, Euclidean space, orthogonal decompositions, matrices, and systems of linear equations.