Principles of Linear Algebra with Mathematica

by Shiskowski, Kenneth M.; Frinkle, Karl

• ISBN13: 9780470637951
• ISBN10: 0470637951
• Edition: 1st
• Format: Hardcover
• Copyright: 2011-08-02
• Publisher: Wiley

Summary

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

Preface	p. ix
Conventions and Notations	p. xiv
An Introduction to Mathematica“	p. 1
The Very Basics	p. 1
Basic Arithmetic	p. 4
Lists and Matrices	p. 9
Expressions versus Functions	p. 12
Plotting and Animations	p. 14
Solving Systems of Equations	p. 24
Basic Programming	p. 28
Linear Systems of Equations and Matrices	p. 31
Linear Systems of Equations	p. 31
Augmented Matrix of a Linear System and Row Operations	p. 44
Some Matrix Arithmetic	p. 54
Gauss-Jordan Elimination and Reduced Row Echelon Form	p. 69
Gauss-Jordan Enmination and rref	p. 69
Elementary Matrices	p. 81
Sensitivity of Solutions to Error in the Linear System	p. 92
Applications of Linear Systems and Matrices	p. 105
Applications of Linear Systems to Geometry	p. 105
Applications of Linear Systems to Curve Fitting	p. 115
Applications of Linear Systems to Economics	p. 122
Applications of Matrix Multiplication to Geometry	p. 127
An Application of Matrix Multiplication to Economics	p. 135
Determinants, Inverses, and Cramer's Rule	p. 143
Determinants and Inverses from the Adjoint Formula	p. 143
Finding Determinants by Expanding along Any Row or Column	p. 161
Determinants Found by Triangularizing Matrices	p. 173
LU Factorization	p. 185
Inverses from rref	p. 192
Gramer's Rule	p. 197
Basic Vector Algebra Topics	p. 207
Vectors	p. 207
Dot Product	p. 221
Cross Product	p. 233
Vector Projection	p. 242
A Few Advanced Vector Algebra Topics	p. 255
Rotations in Space	p. 255
"Rolling" a Circle along a Curve	p. 265
The TNB Frame	p. 275
Independence, Basis, and Dimension for Subspaces of Rn	p. 281
Subspaces of Rn	p. 281
Independent and Dependent Sets of Vectors in Rn	p. 298
Basis and Dimension for Subspaces of Rn	p. 310
Vector Projection onto a Subspace of Rn	p. 320
The Gram-Schmidt Orthonormalization Process	p. 331
Linear Maps from Rn to Rm	p. 341
Basics about Linear Maps	p. 341
The Kernel and Image Subspaces of a Linear Map	p. 353
Composites of Two Linear Maps and Inverses	p. 361
Change of Bases for the Matrix Representation of a Linear Map	p. 368
The Geometry of Linear and Affine Maps
The Effect of a Linear Map on Area and Arclength in Two Dimensions	p. 383
The Decomposition of Linear Maps into Rotations, Reflections, and Rescalings in R2	p. 401
The Effect of Linear Maps on Volume, Area, arid Arclength in R3	p. 409
Rotations, Reflections, and Rescalings in Three Dimensions	p. 421
Affine Maps	p. 431
Least-Squares Fits and Pseudo inverses	p. 443
Pseudoinverse to a Nonsquare Matrix and Almost Solving an Overdetermined Linear System	p. 443
Fits and Pseudoinverses	p. 454
Least-Squares Fits and Pseudoinverses	p. 469
Eigenvalues and Eigenvectors	p. 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 Matrix	p. 496
Applications of the Diagonalizability of Square Matrices	p. 500
Solving a Square First-Order Linear System of Differential Equations	p. 516
Basic Facts about Eigenvalues, Eigenvectors, and Diagonalizability	p. 552
The Geometry of the Ellipse Using Eigenvalues and Eigenvectors	p. 566
A Mathematica Eigen-Function	p. 586
Bibliographic Material	p. 591
Indexes	p. 593
Keyword Index	p. 593
Index of Mathematica Commands	p. 597
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