Principles of Linear Algebra With Maple

by Shiskowski, Kenneth M.; Frinkle, Karl

• ISBN13: 9780470637593
• ISBN10: 0470637595
• Edition: 1st
• Format: Hardcover
• Copyright: 2010-09-28
• Publisher: Wiley

Summary

Linear Algebra with Maple uniquely addresses the quickly growing intersection between subject theory and numerical computation. Computer algebra systems such as Maple are becoming ever more powerful, useful, user friendly and readily available to the average student and professional, but thre are few books which currently cross this gap between linear algebra and Maple. This book introduces algebra topics which can only be taught with the help of computer algebra systems, and the authors include all of the commands required to solve complex and computationally challenging linear algebra problems using Maple. The book begins with an introduction to the commands and programming guidelines for working with Maple. Next, the authors explore linear systems of equations and matrices, applications of linear systems and matrices, determinants, inverses, and Cramer's rule. Basic linear algebra topics, such as vectors, dot product, cross product, vector projection, are explored as well as the more advanced topics of rotations in space, rolling a circle along a curve, and the TNB Frame. Subsequent chapters feature coverage of linear programming, linear transformations from R n to R m , the geometry of linear and affine transformations, and least squares fits and pseudoinverses. Although computational in nature, the material is not presented in a simply theory-proof-problem format. Instead, all topics are explored in a reader-friendly and insightful way. The Maple software is fully utilized to highlight the visual nature of the topic, as the book is complete with numerous graphics in two and three dimensions, animations, symbolic manipulations, numerical computations, and programming. Exercises are supplied in most chapters, and a related Web site houses Maple code so readers can work throug the provided examples.

Author Biography

Kenneth Shiskowski, PhD, is Professor of Mathematics at Eastern Michigan University. His areas of research interest include numerical analysis, the history of mathematics, the integration of technology into mathematics, differential geometry, and dynamical systems.

Karl H. Frinkle, PhD, is Associate Professor of Mathematics at Southeastern Oklahoma State University. He has extensive academic experience teaching in the areas of algebra, trigonometry, and calculus. Dr. Frinkle currently focuses his research on Bose-Einstein condensates, nonlinear optics, dynamical systems, and the integration of technology into mathematics.

Table of Contents

Preface

Conventions and Notations

An Introduction To Maple

The Commands

Programming

Linear Systems of Equations and Matrices

Linear Systems of Equations

Augmented Matrix of a Linear System and Row Operations

Some Matrix Arithmetic

Gauss-Jordan Elimination and Reduced Row Echelon Form

Gauss-Jordan Elimination and rref

Elementary Matrices

Sensitivity of Solutions to Error in the Linear System

Applications of Linear Systems and Matrices

Applications of Linear Systems to Geometry

Applications of Linear Systems to Curve Fitting

Applications of Linear Systems to Economics

Applications of Matrix Multiplication to Geometry

An Application of Matrix Multiplication to Economics

Determinants, Inverses and Cramer's Rule

Determinants and Inverses from the Adjoint Formula

Determinants by Expanding Along Any Row or Column

Determinants Found by Triangularizing Matrices

LU Factorization

Inverses from rref

Cramer's Rule

Basic Linear Algebra Topics

Vectors

Dot Product

Cross Product

Vector Projection

A Few Advanced Linear Algebra Topics

Rotations in Space

'Rolling' a Circle Along a Curve

The TNB Frame

Independence, Basis and Dimension for Subspaces of Rn

Subspaces of Rn

Independent and Dependent Sets of Vectors in Rn

Basis and Dimension for Subspaces of Rn

Vector Projection onto a Subspace of Rn

The Gram-Schmidt Orthonormalization Process

Linear Maps from Rn to Rm

Basics About Linear Maps

The Kernel and Image Subspaces of a Linear Map

Composites of Two Linear maps and Inverses

Change of Bases for the Matrix Representation of a Linear Map

The Geometry of Linear and Affine Maps

The Effect of a Linear Map on Area and Arclength in Two Dimensions

The Decomposition of Linear Maps into Rotations, Reflections and Rescalings in R2

The Effect of Linear Maps on Volume, Area and Arclength in R3

Rotations, Reflections and Rescalings in Three Dimensions

Affine Maps

Least Squares Fits and Pseudoinverses

Pseudoinverse to a Non-Square Matrix and Almost Solving an Overdetermined Linear System

Fits and Pseudoinverses

Least Squares Fits and Pseudoinverses

Eigenvalues and Eigenvectors

What Are Eigenvalues and Eigenvectors, and Why Do We Need Them?

Summary of Definitions and Methods for Computing Eigenvalues and Eigenvectors as well as the Exponential of a Matrix

Applications of the Diagonalizability of Square Matrices

Solving a Square First Order Linear

System of Differential Equations . . . . . . . . . . . . . . . .

Basic Facts About Eigenvalues and Eigenvectors, and Diagonalizability

The Geometry of the Ellipse Using Eigenvalues and Eigenvectors

A Maple Eigen-Procedure

Bibliography

Indices

Keyword Index

Index of Maple Commands and Packages

Table of Contents provided by Publisher. All Rights Reserved.