Principles of Linear Algebra With Maple

Linear Algebra with Maple (R) uniquely addresses the quickly growing intersection between subject theory and numerical computation. Computer algebra systems such as Maple 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 Maple. 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 Maple. The book begins with an introduction to the commands and programming guidelines for working with Maple. 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 Rm, 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 Maple 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 Maple code so readers can work throug the provided examples.

Spis treści:
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.