Principles Of Linear Algebra With Mathematica (R)

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

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. Dr. Shiskowski is the coauthor of Principles of Linear Algebra with Maple, published by Wiley.

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. Dr. Frinkle is the coauthor of Principles of Linear Algebra with Maple, published by Wiley.

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 Echelon Form.

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 of R_n.

8.1 Subspaces of R_n.

8.2 Independent and Dependent Sets of Vectors in R_n.

8.3 Basis and Dimension for Subspaces of R_n.

8.4 Vector Projection onto a subspace of R_n.

8.5 The Gram-Schmidt Orthonormalization Process.

9. Linear Maps from R_n to R_m.

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 Linear Map.

10. The Geometry of Linear and Affine Maps.

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

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 in R₃.

10.4 Rotations, Reflections, and Rescalings in Three Dimensions.

10.5 Affine Maps.

11. Least-Squares Fits and Pseudoinverses.

11.1 Pseudoinverse to a Nonsquare Matrix and Almost Solving an Overdetermined 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 Need Them?

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

12.3 Applications of the Diagonalizability of Square Matrices.

12.4 Solving a Square First-Order Linear System if Differential Equations.

12.5 Basic Facts About Eigenvalues, Eigenvectors, and Diagonalizability.

12.6 The Geometry of the Ellipse Using Eigenvalues and Eigenvectors.

12.7 A Mathematica EigenFunction.

Suggested Reading.

Indices.

Keyword Index.

Index of Mathematica Commands.