Linear Algebra: A Modern Introduction

✍ Scribed by David Poole

• Publisher: Cengage Learning
• Year: 2014
• Tongue: English
• Leaves: 752
• Edition: 4
• Category: Library

✦ Synopsis

David Poole's innovative LINEAR ALGEBRA: A MODERN INTRODUCTION, 4e emphasizes a vectors approach and better prepares students to make the transition from computational to theoretical mathematics. Balancing theory and applications, the book is written in a conversational style and combines a traditional presentation with a focus on student-centered learning. Theoretical, computational, and applied topics are presented in a flexible yet integrated way. Stressing geometric understanding before computational techniques, vectors and vector geometry are introduced early to help students visualize concepts and develop mathematical maturity for abstract thinking. Additionally, the book includes ample applications drawn from a variety of disciplines, which reinforce the fact that linear algebra is a valuable tool for modeling real-life problems.

✦ Table of Contents

Front Cover
Table of Contents
Preface
To the Instructor
To the student
Chapter 1 - Vectors - pg. 1
1.0 - Introduction: The Racetrack Game - pg. 1
1.1 - The Geometry and Algebra of Vectors - pg. 3
1.2 - Length and Angle: The Dot Product - pg. 18
Exploration: Vectors and Geometry - pg. 32
1.3 - Lines and Planes - pg. 34
Exploration: The Cross Product - pg. 48
Writing Project: The Origins of the Dot Product and Cross Product - pg. 49
1.4 - Applications - pg. 50
Force Vectors - pg. 50
Chapter Review - pg. 55
Chapter 2 - Systems of Linear Equations - pg. 57
2.0 - Introduction: Triviality - pg. 57
2.1 - Introduction to Systems of Linear Equations - pg. 58
2.2 - Direct Methods for Solving Linear Systems - pg. 64
Writing Project: A History of Gaussian Elimination - pg. 82
Explorations: Lies My Computer Told Me - pg. 83
Partial Pivoting - pg. 84
Counting Operations: An Introduction to the Analysis of Algorithms - pg. 85
2.3 - Spanning Sets and Linear Independence - pg. 88
2.4 - Applications - pg. 99
Allocation of Resources - pg. 99
Balancing Chemical Equations - pg. 101
Network Analysis - pg. 102
Electrical Networks - pg. 104
Linear Economic Models - pg. 107
Finite Linear Games - pg. 109
Vignette: The Global Positioning System - pg. 121
2.5 - Iterative Methods for Solving Linear Systems - pg. 124
Chapter Review - pg. 134
Chapter 3 - Matrices - pg. 136
3 - Introduction: Matrices in Action - pg. 136
3.1 - Matrix Operations - pg. 138
3.2 - Matrix Algebra - pg. 154
3.3 - The Inverse of a Matrix - pg. 163
3.4 - The LU Factorization - pg. 180
3.5 - Subspaces, Basis, Dimension, and Rank - pg. 191
3.6 - Introduction to Linear Transformations - pg. 211
Vignette: Robotics - pg. 226
3.7 - Applications - pg. 230
Markov Chains - pg. 230
Linear Economic Models - pg. 235
Population Growth - pg. 239
Graphs and Digraphs - pg. 241
Chapter Review - pg. 251
Chapter 4 - Eigenvalues and Eigenvectors - pg. 253
4 - Introduction: A Dynamical System on Graphs - pg. 253
4.1 - Introduction to Eigenvalues and Eigenvectors - pg. 254
4.2 - Determinants - pg. 263
Writing Project: Which Came First: The Matrix or the Determinant? - pg. 283
Vignette: Lewis Carroll's Condensation Method - pg. 284
Exploration: Geometric Applications of Determinants - pg. 286
4.3 - Eigenvalues and Eigenvectors of n X n Matrices - pg. 292
4.4 - Similarity and Diagonalization - pg. 301
4.5 - Iterative Methods for Computing Eigenvalues - pg. 311
4.6 - Applications and the Perron-Frobenius Theorem - pg. 325
Markov Chains - pg. 325
Population Growth - pg. 330
The Perron-Frobenius Theorem - pg. 332
Linear Recurrence Relations - pg. 335
Systems of Linear Differential Equations - pg. 340
Discrete Linear Dynamical Systems - pg. 348
Vignette: Ranking Sports Teams and Searching the Internet - pg. 356
Chapter Review - pg. 364
Chapter 5 - Orthogonality - pg. 366
5.0 - Introduction: Shadows on a Wall - pg. 366
5.1 - Orthogonality in IR" - pg. 368
5.2 - Orthogonal Complements and Orthogonal Projections - pg. 378
5.3 - The Gram-Schmidt Process and the QR Factorization - pg. 388
Explorations: The Modified QR Factorization - pg. 396
Approximating Eigenvalues with the QR Algorithm - pg. 398
5.4 - Orthogonal Diagonalization of Symmetric Matrices - pg. 400
5.5 - Applications - pg. 408
Quadratic Forms - pg. 408
Graphing Quadratic Equations - pg. 415
Chapter Review - pg. 425
Chapter 6 - Vector Spaces - pg. 427
6 - Introduction: Fibonacci in (Vector) Space - pg. 427
6.1 - Vector Spaces and Subspaces - pg. 429
Writing Project: The Rise of Vector Spaces - pg. 443
6.2 - Linear Independence, Basis, and Dimension - pg. 443
Exploration: Magic Squares - pg. 460
6.3 - Change of Basis - pg. 463
6.4 - Linear Transformations - pg. 472
6.5 - The Kernel and Range of a Linear Transformation - pg. 481
6.6 - The Matrix of a Linear Transformation - pg. 497
Exploration: Tilings, Lattices, and the Crystallographic Restriction - pg. 515
6.7 - Applications - pg. 518
Homogeneous Linear Differential Equations - pg. 518
Chapter Review - pg. 527
Chapter 7 - Distance and Approximation - pg. 529
7.0 - Introduction: Taxicab Geometry - pg. 529
7.1 - Inner Product Spaces - pg. 531
Explorations: Vectors and Matrices with Complex Entries - pg. 543
Geometric Inequalities and Optimization Problems - pg. 547
7.2 - Norms and Distance Functions - pg. 552
7.3 - Least Squares Approximation - pg. 568
7.4 - The Singular Value Decomposition - pg. 590
Vignette: Digital Image Compression - pg. 607
7.5 - Applications - pg. 610
Approximation of Functions - pg. 610
Chapter Review - pg. 618
Chapter 8 - Codes Online only - pg. 620
8.1 - Code Vectors - pg. 620
Vignette: The Codabar System - pg. 626
8.2 - Error-Correcting Codes - pg. 627
8.3 - Dual Codes - pg. 632
8.4 - Linear Codes - pg. 639
8.5 - The Minimum Distance of a Code - pg. 644