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Principles of Linear Algebra With Maple

ISBN: 978-0-470-63759-3
596 pages
September 2010
Principles of Linear Algebra With Maple (0470637595) cover image
An accessible introduction to the theoretical and computational aspects of linear algebra using MapleTM

Many topics in linear algebra can be computationally intensive, and software programs often serve as important tools for understanding challenging concepts and visualizing the geometric aspects of the subject. Principles of Linear Algebra with Maple uniquely addresses the quickly growing intersection between subject theory and numerical computation, providing all of the commands required to solve complex and computationally challenging linear algebra problems using Maple. The authors supply an informal, accessible, and easy-to-follow treatment of key topics often found in a first course in linear algebra.

Requiring no prior knowledge of the software, the book begins with an introduction to the commands and programming guidelines for working with Maple. Next, the book explores 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, and vector projection are explained, 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 transformations from Rn to Rm, the geometry of linear and affine transformations, least squares fits and pseudoinverses, and eigenvalues and eigenvectors.

The authors explore several topics that are not often found in introductory linear algebra books, including sensitivity to error and the effects of linear and affine maps on the geometry of objects. The Maple software highlights the topic's visual nature, as the book is complete with numerous graphics in two and three dimensions, animations, symbolic manipulations, numerical computations, and programming. In addition, a related Web site features supplemental material, including Maple code for each chapter's problems, solutions, and color versions of the book's figures.

Extensively class-tested to ensure an accessible presentation, Principles of Linear Algebra with Maple is an excellent book for courses on linear algebra at the undergraduate level. It is also an ideal reference for students and professionals who would like to gain a further understanding of the use of Maple to solve linear algebra problems.

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Preface.

Conventions and Notations.

1 An Introduction To Maple.

1.1 The Commands .

1.2 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’s 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 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 Non-Square 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 of Differential Equations . . . . . . . . . . . . . . . . . .

12.5 Basic Facts About Eigenvalues and Eigenvectors, and Diagonalizability.

12.6 The Geometry of the Ellipse Using Eigenvalues and Eigenvectors.

12.7 A Maple Eigen-Procedure.

Bibliography.

Indices.

Keyword Index.

Index of Maple Commands and Packages.

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Kenneth Shiskowski, PhD, is Professor of Mathematics at Eastern Michigan University. His areas of research interest include numerical analysis, the history of mathematics, the integration of technology into mathematics, differential geometry, and dynamical systems.

Karl H. Frinkle, PhD, is Associate Professor of Mathematics at Southeastern Oklahoma State University. He has extensive academic experience teaching in the areas of algebra, trigonometry, and calculus. Dr. Frinkle currently focuses his research on Bose-Einstein condensates, nonlinear optics, dynamical systems, and the integration of technology into mathematics.

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