Linear Algebra A Modern Introduction,9780534341749

Linear Algebra A Modern Introduction

by
ISBN13: 9780534341749
ISBN10: 0534341748
Format: Hardcover
Pub. Date: 2002-03-13
Publisher(s): Brooks Cole
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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

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