Linear algebra: a modern introduction

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