Principles of Linear Algebra with Mathematica (eBook)

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 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.

"An accessible introduction to the theoretical and computational aspects of linear algebra using Maple(TM)." (TMCnet.com, 16 April 2011)