Principles of Linear Algebra with Mathematica

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  • Format: Hardcover
  • Copyright: 2011-08-02
  • Publisher: Wiley

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Principles of Linear Algebra with Mathematicar uniquely addresses the quickly growing intersection between subject theory and numerical computation. Computer algebra systems such as Mathematicar 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 Mathematicar. 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 Mathematicar. The book begins with an introduction to the commands and programming guidelines for working with Mathematicar. 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 Rn to Rm, 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 Mathematicar 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 Mathematicar code so readers can work through the provided examples.

Author Biography

Kenneth Shiskowski, PHD, is Professor of Mathematics at Eastern Michigan University. His areas of research interest include numerical analysis, history of mathematics, the integration of technology into mathematics, differential geometry, and dynamical systems. Karl Frinkle, PHD, is Associate Professor of Mathematics at Southeastern Oklahoma State University. His areas of research include Bose-Einstein condensates, nonlinear optics, dynamical systems, and integrating technology into mathematics.

Table of Contents

Prefacep. ix
Conventions and Notationsp. xiv
An Introduction to Mathematica“p. 1
The Very Basicsp. 1
Basic Arithmeticp. 4
Lists and Matricesp. 9
Expressions versus Functionsp. 12
Plotting and Animationsp. 14
Solving Systems of Equationsp. 24
Basic Programmingp. 28
Linear Systems of Equations and Matricesp. 31
Linear Systems of Equationsp. 31
Augmented Matrix of a Linear System and Row Operationsp. 44
Some Matrix Arithmeticp. 54
Gauss-Jordan Elimination and Reduced Row Echelon Formp. 69
Gauss-Jordan Enmination and rrefp. 69
Elementary Matricesp. 81
Sensitivity of Solutions to Error in the Linear Systemp. 92
Applications of Linear Systems and Matricesp. 105
Applications of Linear Systems to Geometryp. 105
Applications of Linear Systems to Curve Fittingp. 115
Applications of Linear Systems to Economicsp. 122
Applications of Matrix Multiplication to Geometryp. 127
An Application of Matrix Multiplication to Economicsp. 135
Determinants, Inverses, and Cramer's Rulep. 143
Determinants and Inverses from the Adjoint Formulap. 143
Finding Determinants by Expanding along Any Row or Columnp. 161
Determinants Found by Triangularizing Matricesp. 173
LU Factorizationp. 185
Inverses from rrefp. 192
Gramer's Rulep. 197
Basic Vector Algebra Topicsp. 207
Vectorsp. 207
Dot Productp. 221
Cross Productp. 233
Vector Projectionp. 242
A Few Advanced Vector Algebra Topicsp. 255
Rotations in Spacep. 255
"Rolling" a Circle along a Curvep. 265
The TNB Framep. 275
Independence, Basis, and Dimension for Subspaces of Rnp. 281
Subspaces of Rnp. 281
Independent and Dependent Sets of Vectors in Rnp. 298
Basis and Dimension for Subspaces of Rnp. 310
Vector Projection onto a Subspace of Rnp. 320
The Gram-Schmidt Orthonormalization Processp. 331
Linear Maps from Rn to Rmp. 341
Basics about Linear Mapsp. 341
The Kernel and Image Subspaces of a Linear Mapp. 353
Composites of Two Linear Maps and Inversesp. 361
Change of Bases for the Matrix Representation of a Linear Mapp. 368
The Geometry of Linear and Affine Maps
The Effect of a Linear Map on Area and Arclength in Two Dimensionsp. 383
The Decomposition of Linear Maps into Rotations, Reflections, and Rescalings in R2p. 401
The Effect of Linear Maps on Volume, Area, arid Arclength in R3p. 409
Rotations, Reflections, and Rescalings in Three Dimensionsp. 421
Affine Mapsp. 431
Least-Squares Fits and Pseudo inversesp. 443
Pseudoinverse to a Nonsquare Matrix and Almost Solving an Overdetermined Linear Systemp. 443
Fits and Pseudoinversesp. 454
Least-Squares Fits and Pseudoinversesp. 469
Eigenvalues and Eigenvectorsp. 481
What Are Eigenvalues and Eigenvectors, and Why Do We Need Them?p. 481
Summary of Definitions and Methods for Computing Eigenvalues and Eigenvectors as Well as the Exponential of a Matrixp. 496
Applications of the Diagonalizability of Square Matricesp. 500
Solving a Square First-Order Linear System of Differential Equationsp. 516
Basic Facts about Eigenvalues, Eigenvectors, and Diagonalizabilityp. 552
The Geometry of the Ellipse Using Eigenvalues and Eigenvectorsp. 566
A Mathematica Eigen-Functionp. 586
Bibliographic Materialp. 591
Indexesp. 593
Keyword Indexp. 593
Index of Mathematica Commandsp. 597
Table of Contents provided by Ingram. All Rights Reserved.

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