Linear Algebra A Modern Introduction (with CD-ROM)

by Poole, David

• ISBN13: 9780534998455
• ISBN10: 0534998453
• Edition: 2nd
• Format: Hardcover
• Copyright: 2005-01-24
• Publisher: Brooks Cole

Summary

David Poole's innovative book emphasizes vectors and geometric intuition from the start and better prepares students to make the transition from the computational aspects of the course to the theoretical. Poole covers vectors and vector geometry first to enable students to visualize the mathematics while they are doing matrix operations. With a concrete understanding of vector geometry, students are able to visualize and understand the meaning of the calculations that they will encounter. By seeing the mathematics and understanding the underlying geometry, students develop mathematical maturity and can think abstractly when they reach vector spaces. Throughout the text, Poole's direct conversational writing style connects with students, and an abundant selection of applications from a broad range of disciplines clearly demonstrates the relevance of linear algebra.

Table of Contents

Preface	vii
To the Instructor	xvii
To the Student	xxiii
Vectors	1	(57)
Introduction: The Racetrack Game	1	(2)
The Geometry and Algebra of Vectors	3	(12)
Length and Angle: The Dot Product	15	(16)
Exploration: Vectors and Geometry	29	(2)
Lines and Planes	31	(16)
Exploration: The Cross Product	45	(2)
Code Vectors and Modular Arithmetic	47	(11)
Vignette: The Codabar System	55	(1)
Chapter Review	56	(2)
Systems of Linear Equations	58	(76)
Introduction: Triviality	58	(1)
Introduction to Systems of Linear Equations	59	(9)
Exploration: Lies My Computer Told Me	66	(2)
Direct Methods for Solving Linear Systems	68	(22)
Exploration: Partial Pivoting	86	(1)
Exploration: Counting Operations: An Introduction to the Analysis of Algorithms	87	(3)
Spanning Sets and Linear Independence	90	(11)
Applications	101	(21)
Allocation of Resources	101	(2)
Balancing Chemical Equations	103	(1)
Network Analysis	104	(2)
Electrical Networks	106	(3)
Finite Linear Games	109	(10)
Vignette: The Global Positioning System	119	(3)
Iterative Methods for Solving Linear Systems	122	(12)
Chapter Review	132	(2)
Matrices	134	(118)
Introduction: Matrices in Action	134	(2)
Matrix Operations	136	(16)
Matrix Algebra	152	(9)
The Inverse of a Matrix	161	(17)
The LU Factorization	178	(11)
Subspaces, Basis, Dimension, and Rank	189	(20)
Introduction to Linear Transformations	209	(19)
Vignette: Robotics	224	(4)
Applications	228	(24)
Markov Chains	228	(5)
Population Growth	233	(2)
Graphs and Digraphs	235	(5)
Error-Correcting Codes	240	(10)
Chapter Review	250	(2)
Eigenvalues and Eigenvectors	252	(111)
Introduction: A Dynamical System on Graphs	252	(1)
Introduction to Eigenvalues and Eigenvectors	253	(9)
Determinants	262	(27)
Exploration: Geometric Applications of Determinants	283	(6)
Eigenvalues and Eigenvectors of nxn Matrices	289	(9)
Similarity and Diagonalization	298	(10)
Iterative Methods for Computing Eigenvalues	308	(14)
Applications and the Perron-Frobenius Theorem	322	(41)
Markov Chains	322	(5)
Population Growth	327	(2)
The Perron-Frobenius Theorem	329	(3)
Linear Recurrence Relations	332	(5)
Systems of Linear Differential Equations	337	(8)
Discrete Linear Dynamical Systems	345	(8)
Vignette: Ranking Sports Teams and Searching the Internet	353	(8)
Chapter Review	361	(2)
Orthogonality	363	(68)
Introduction: Shadows on a Wall	363	(2)
Orthogonality in Rn	365	(10)
Orthogonal Complements and Orthogonal Projections	375	(10)
The Gram-Schmidt Process and the QR Factorization	385	(12)
Exploration: The Modified QR Factorization	393	(2)
Exploration: Approximating Eigenvalues with the QR Algorithm	395	(2)
Orthogonal Diagonalization of Symmetric Matrices	397	(8)
Applications	405	(26)
Dual Codes	405	(6)
Quadratic Forms	411	(7)
Graphing Quadratic Equations	418	(11)
Chapter Review	429	(2)
Vector Spaces	431	(107)
Introduction: Fibonacci in (Vector) Space	431	(2)
Vector Spaces and Subspaces	433	(14)
Linear Independence, Basis, and Dimension	447	(20)
Exploration: Magic Squares	464	(3)
Change of Basis	467	(9)
Linear Transformations	476	(9)
The Kernel and Range of a Linear Transformation	485	(16)
The Matrix of a Linear Transformation	501	(21)
Exploration: Tilings, Lattices, and the Crystallographic Restriction	519	(3)
Applications	522	(16)
Homogeneous Linear Differential Equations	522	(7)
Linear Codes	529	(7)
Chapter Review	536	(2)
Distance and Approximation	538	(96)
Introduction: Taxicab Geometry	538	(2)
Inner Product Spaces	540	(21)
Exploration: Vectors and Matrices with Complex Entries	552	(4)
Exploration: Geometric Inequalities and Optimization Problems	556	(5)
Norms and Distance Functions	561	(16)
Least Squares Approximation	577	(22)
The Singular Value Decomposition	599	(20)
Vignette: Digital Image Compression	616	(3)
Applications	619	(15)
Approximation of Functions	619	(7)
Error-Correcting Codes	626	(5)
Chapter Review	631	(3)
Appendix A Mathematical Notation and Methods of Proof	634	(9)
Appendix B Mathematical Induction	643	(7)
Appendix C Complex Numbers	650	(11)
Appendix D Polynomials	661	(10)
Answers to Selected Odd-Numbered Exercises	671	(35)
Index	706