Principles of Linear Algebra with Mathematica
Published on 7. June 2013
616 pages
978-1-118-62726-6 (ISBN)
Description
A hands-on introduction to the theoretical and computationalaspects of linear algebra using Mathematica®
Many topics in linear algebra are simple, yet computationallyintensive, and computer algebra systems such as Mathematica®are essential not only for learning to apply the concepts tocomputationally challenging problems, but also for visualizing manyof the geometric aspects within this field of study. Principles ofLinear Algebra with Mathematica uniquely bridges the gap betweenbeginning linear algebra and computational linear algebra that isoften encountered in applied settings, and the commands required tosolve complex and computationally challenging problems usingMathematica are provided.
The book begins with an introduction to the commands andprogramming guidelines for working with Mathematica. Next, theauthors explore linear systems of equations and matrices,applications of linear systems and matrices, determinants,inverses, and Cramer's rule. Basic linear algebra topics, such asvectors, dot product, cross product, and vector projection areexplored, as well as a unique variety of more advanced topicsincluding rotations in space, 'rolling' a circle along a curve, andthe TNB Frame. Subsequent chapters feature coverage of lineartransformations from Rn to Rm, the geometry of linear and affinetransformations, with an exploration of their effect on arclength,area, and volume, least squares fits, and pseudoinverses.
Mathematica is used to enhance concepts and is seamlesslyintegrated throughout the book through symbolic manipulations,numerical computations, graphics in two and three dimensions,animations, and programming. Each section concludes with standardproblems in addition to problems that were specifically designed tobe solved with Mathematica, allowing readers to test theircomprehension of the presented material. All related Mathematicacode is available on a corresponding website, along with solutionsto problems and additional topical resources.
Extensively class-tested to ensure an accessible presentation,Principles of Linear Algebra with Mathematica is an excellent bookfor courses on linear algebra at the undergraduate level. The bookis also an ideal reference for students and professionals who wouldlike to gain a further understanding of the use of Mathematica tosolve linear algebra problems.
Reviews / Votes
"An accessible introduction to the theoretical and computational aspects of linear algebra using Maple(TM)." (TMCnet.com, 16 April 2011)More details
Other editions
Additional editions
Kenneth M. Shiskowski | Karl Frinkle
Principles of Linear Algebra with Mathematica (R)
Book
07/2011
Wiley
€115.00
Shipment within 10-20 days
Persons
Kenneth Shiskowski, PhD, is Professor of Mathematics at EasternMichigan University. His areas of research interest includenumerical analysis, history of mathematics, the integration oftechnology into mathematics, differential geometry, and dynamicalsystems. Dr. Shiskowski is the coauthor of Principles of LinearAlgebra with Maple, published by Wiley.
Karl Frinkle, PhD, is Associate Professor of Mathematics atSoutheastern Oklahoma State University. His areas of researchinclude Bose-Einstein condensates, nonlinear optics, dynamicalsystems, and integrating technology into mathematics. Dr. Frinkleis the coauthor of Principles of Linear Algebra with Maple,published by Wiley.
Content
Preface.
Conventions and Notations.
1. An Introduction to Mathematica.
1.1 The Very Basics.
1.2 Basic Arithmetic.
1.3 Lists and Matrices.
1.4 Expressions Versus Functions.
1.5 Plotting and Animations.
1.6 Solving Systems of Equations.
1.7 Basic Programming.
2. Linear Systems of Equations and Matrices.
2.1 Linear Systems of Equations.
2.2 Augmented Matrix of a Linear System and Row Operations.
2.3 Some Matrix Arithmetic.
3. Gauss-Jordan Elimination and Reduced Row EchelonForm.
3.1 Gauss-Jordan Elimination and rref.
3.2 Elementary Matrices.
3.3 Sensitivity of Solutions to Error in the Linear System.
4. Applications of Linear Systems and Matrices.
4.1 Applications of Linear Systems to Geometry.
4.2 Applications of Linear Systems to Curve Fitting.
4.3 Applications of Linear Systems to Economics.
4.4 Applications of Matrix Multiplication to Geometry.
4.5 An Application of Matrix Multiplication to Economics.
5. Determinants, Inverses, and Cramer' Rule.
5.1 Determinants and Inverses from the Adjoint Formula.
5.2 Determinants by Expanding Along Any Row or Column.
5.3 Determinants Found by Triangularizing Matrices.
5.4 LU Factorization.
5.5 Inverses from rref.
5.6 Cramer's Rule.
6. Basic Linear Algebra Topics.
6.1 Vectors.
6.2 Dot Product.
6.3 Cross Product.
6.4 A Vector Projection.
7. A Few Advanced Linear Algebra Topics.
7.1 Rotations in Space.
7.2 "Rolling" a Circle Along a Curve.
7.3 The TNB Frame.
8. Independence, Basis, and Dimension for Subspaces ofRn.
8.1 Subspaces of Rn.
8.2 Independent and Dependent Sets of Vectors inRn.
8.3 Basis and Dimension for Subspaces of Rn.
8.4 Vector Projection onto a subspace of Rn.
8.5 The Gram-Schmidt Orthonormalization Process.
9. Linear Maps from Rn to Rm.
9.1 Basics About Linear Maps.
9.2 The Kernel and Image Subspaces of a Linear Map.
9.3 Composites of Two Linear Maps and Inverses.
9.4 Change of Bases for the Matrix Representation of a LinearMap.
10. The Geometry of Linear and Affine Maps.
10.1 The Effect of a Linear Map on Area and Arclength in TwoDimensions.
10.2 The Decomposition of Linear Maps into Rotations,Reflections, and Rescalings in R2.
10.3 The Effect of Linear Maps on Volume, Area, and Arclength inR3.
10.4 Rotations, Reflections, and Rescalings in ThreeDimensions.
10.5 Affine Maps.
11. Least-Squares Fits and Pseudoinverses.
11.1 Pseudoinverse to a Nonsquare Matrix and Almost Solving anOverdetermined Linear System.
11.2 Fits and Pseudoinverses.
11.3 Least-Squares Fits and Pseudoinverses.
12. Eigenvalues and Eigenvectors.
12.1 What Are Eigenvalues and Eigenvectors, and Why Do We NeedThem?
12.2 Summary of Definitions and Methods for ComputingEigenvalues and Eigenvectors as well as the Exponential of aMatrix.
12.3 Applications of the Diagonalizability of SquareMatrices.
12.4 Solving a Square First-Order Linear System if DifferentialEquations.
12.5 Basic Facts About Eigenvalues, Eigenvectors, andDiagonalizability.
12.6 The Geometry of the Ellipse Using Eigenvalues andEigenvectors.
12.7 A Mathematica EigenFunction.
Suggested Reading.
Indices.
Keyword Index.
Index of Mathematica Commands.
