Principles of Linear Algebra with Mathematica (R) (Pure and Applied Mathematics


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Table of Contents


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 Rn.

8.1 Subspaces of Rn.

8.2 Independent and Dependent Sets of Vectors in Rn.

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 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.


Keyword Index.

Index of Mathematica Commands. 

About the Author

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.


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

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