Linear Algebra A Modern Introduction

by Poole, David

• ISBN13: 9780534341749
• ISBN10: 0534341748
• Format: Hardcover
• Copyright: 2002-03-13
• Publisher: Brooks Cole

Summary

In this innovative new Linear Algebra text, award-winning educator David Poole covers vectors and vector geometry first to enable students to visualize the mathematics while they are doing matrix operations. Rather than merely doing the calculations with no understanding of the mathematics, students will be able to visualize and understand the meaning of the calculations. By seeing the mathematics and understanding the underlying geometry, students will develop mathematical maturity and learn to think abstractly.

Table of Contents

Preface	xi
To the Instructor	xix
To the Student	xxv
Vectors	1	(56)
Introduction: The Racetrack Game	1	(2)
The Geometry and Algebra of Vectors	3	(13)
Length and Angle: The Dot Product	16	(16)
Exploration: Vectors and Geometry	30	(2)
Lines and Planes	32	(16)
Exploration: The Cross Product	46	(2)
Code Vectors and Modular Arithmetic	48	(9)
Systems of Linear Equations	57	(74)
Introduction: Triviality	57	(1)
Introduction to Systems of Linear Equations	58	(9)
Exploration: Lies My Computer Told Me	65	(2)
Direct Methods for Solving Linear Systems	67	(22)
Exploration: Partial Pivoting	85	(1)
Exploration: Counting Operations---An Introduction to the Analysis of Algorithms	86	(3)
Spanning Sets and Linear Independence	89	(12)
Applications	101	(19)
Iterative Methods for Solving Linear Systems	120	(11)
Matrices	131	(116)
Introduction: Matrices in Action	131	(2)
Matrix Operations	133	(17)
Matrix Algebra	150	(10)
The Inverse of a Matrix	160	(19)
Subspaces, Basis, Dimension, and Rank	179	(22)
Introduction to Linear Transformations	201	(16)
Applications	217	(30)
Exploration: The LU Factorization	241	(6)
Eigenvalues and Eigenvectors	247	(103)
Introduction: A Dynamical System on Graphs	247	(2)
Introduction to Eigenvalues and Eigenvectors	249	(7)
Determinants	256	(30)
Exploration: Geometric Applications of Determinants	280	(6)
Eigenvalues and Eigenvectors of nXn Matrices	286	(10)
Similarity and Diagonalization	296	(11)
Iterative Methods for Computing Eigenvalues	307	(11)
Applications and the Perron-Frobenius Theorem	318	(32)
Orthogonality	350	(75)
Introduction: Shadows on a Wall	350	(3)
Orthogonality in Rn	353	(11)
Orthogonal Complements and Orthogonal Projections	364	(11)
The Gram-Schmidt Process and the QR Factorization	375	(14)
Exploration: The Modified QR Factorization	384	(3)
Exploration: Approximating Eigenvalues with the QR Algorithm	387	(2)
Orthogonal Diagonalization of Symmetric Matrices	389	(9)
Applications	398	(27)
Vector Spaces	425	(112)
Introduction: Magic Squares	425	(2)
Vector Spaces and Subspaces	427	(16)
Linear Independence, Basis, and Dimension	443	(16)
Change of Basis	459	(11)
Linear Transformations	470	(10)
The Kernel and Range of a Linear Transformation	480	(16)
The Matrix of a Linear Transformation	496	(24)
Exploration: Tilings, Lattices, and the Crystallographic Restriction	517	(3)
Applications	520	(17)
Distance and Approximation	537	(98)
Introduction: Taxicab Geometry	537	(2)
Inner Product Spaces	539	(18)
Exploration: Geometric Inequalities and Optimization Problems	552	(5)
Norms and Distance Functions	557	(18)
Least Squares Approximation	575	(23)
The Singular Value Decomposition	598	(22)
Applications	620	(15)
Appendix A Mathematical Notation and Methods of Proof	635	(10)
Appendix B Mathematical Induction	645	(8)
Appendix C Complex Numbers	653	(12)
Appendix D Polynomials	665	(12)
Appendix E Technology Bytes	677	(46)
Answers to Selected Odd-Numbered Exercises	723	(33)
Index	756