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Table of Contents
Intro
Introduction
Advice for Instructors
Acknowledgments
Contents
1 Vectors
1.1 Coordinates and Vectors
1.2 The Vector Norm
1.3 Angles and the Inner Product
1.4 Inner Product and Vector Arithmetic
1.5 Statistical Correlation
1.6 Information Retrieval
1.6.1 Comparing Movie Viewers
1.7 Distance on a Sphere
1.8 Bézier Curves
1.9 Orthogonal Vectors
1.10 Area of a Parallelogram
1.11 Projection and Reflection
1.12 The ``All-1s'' Vector
1.13 Exercises
1.14 Projects
2 Matrices
2.1 Matrices
2.1.1 Algebraic Properties of Matrix Arithmetic
2.2 Matrix Multiplication
2.2.1 Algebraic Properties of Matrix Multiplication
2.3 The Identity Matrix, I
2.4 Matrix Inverses
2.5 Transpose of a Matrix
2.6 Exercises
3 Matrix Contexts
3.1 Digital Images
3.2 Information Retrieval Revisited
3.3 Markov Processes: A First Look
3.4 Graphs and Networks
3.5 Simple Linear Regression
3.6 k-Means
3.7 Projection and Reflection Revisited
3.8 Geometry of 22 Matrices
3.9 The Matrix Exponential
3.10 Exercises
3.11 Projects
4 Linear Systems
4.1 Linear Equations
4.2 Systems of Linear Equations
4.3 Row Reduction
4.4 Row Echelon Forms
4.5 Matrix Inverses (And How to Find Them)
4.6 Leontief Input-Output Matrices
4.7 Cubic Splines
4.8 Solutions to AX=B
4.9 LU Decomposition
4.10 Affine Projections
4.10.1 Kaczmarz's Method
4.10.2 Fixed Point of an Affine Transformation
4.11 Exercises
4.12 Projects
5 Least Squares and Matrix Geometry
5.1 The Column Space of a Matrix
5.2 Least Squares: Projection into Col(A)
5.3 Least Squares: Two Applications
5.3.1 Multiple Linear Regression
5.3.2 Curve Fitting with Least Squares
5.4 Four Fundamental Subspaces
5.4.1 Column-Row Factorization
5.5 Geometry of Transformations
5.6 Matrix Norms
5.7 Exercises
5.8 Project
6 Orthogonal Systems
6.1 Projections Revisited
6.2 Building Orthogonal Sets
6.3 QR Factorization
6.4 Least Squares with QR
6.5 Orthogonality and Matrix Norms
6.6 Exercises
6.7 Projects
7 Eigenvalues
7.1 Eigenvalues and Eigenvectors
7.2 Computing Eigenvalues
7.3 Computing Eigenvectors
7.4 Transformation of Eigenvalues
7.5 Eigenvalue Decomposition
7.6 Population Models
7.7 Rotations of R3
7.8 Existence of Eigenvalues
7.9 Exercises
8 Markov Processes
8.1 Stochastic Matrices
8.2 Stationary Distributions
8.3 The Power Method
8.4 Two-State Markov Processes
8.5 Ranking Web Pages
8.6 The Monte Carlo Method
8.7 Random Walks on Graphs
8.8 Exercises
8.9 Project
9 Symmetric Matrices
9.1 The Spectral Theorem
9.2 Norm of a Symmetric Matrix
9.3 Positive Semidefinite Matrices
9.3.1 Matrix Square Roots
9.4 Clusters in a Graph
9.5 Clustering a Graph with k-Means
9.6 Drawing a Graph
Introduction
Advice for Instructors
Acknowledgments
Contents
1 Vectors
1.1 Coordinates and Vectors
1.2 The Vector Norm
1.3 Angles and the Inner Product
1.4 Inner Product and Vector Arithmetic
1.5 Statistical Correlation
1.6 Information Retrieval
1.6.1 Comparing Movie Viewers
1.7 Distance on a Sphere
1.8 Bézier Curves
1.9 Orthogonal Vectors
1.10 Area of a Parallelogram
1.11 Projection and Reflection
1.12 The ``All-1s'' Vector
1.13 Exercises
1.14 Projects
2 Matrices
2.1 Matrices
2.1.1 Algebraic Properties of Matrix Arithmetic
2.2 Matrix Multiplication
2.2.1 Algebraic Properties of Matrix Multiplication
2.3 The Identity Matrix, I
2.4 Matrix Inverses
2.5 Transpose of a Matrix
2.6 Exercises
3 Matrix Contexts
3.1 Digital Images
3.2 Information Retrieval Revisited
3.3 Markov Processes: A First Look
3.4 Graphs and Networks
3.5 Simple Linear Regression
3.6 k-Means
3.7 Projection and Reflection Revisited
3.8 Geometry of 22 Matrices
3.9 The Matrix Exponential
3.10 Exercises
3.11 Projects
4 Linear Systems
4.1 Linear Equations
4.2 Systems of Linear Equations
4.3 Row Reduction
4.4 Row Echelon Forms
4.5 Matrix Inverses (And How to Find Them)
4.6 Leontief Input-Output Matrices
4.7 Cubic Splines
4.8 Solutions to AX=B
4.9 LU Decomposition
4.10 Affine Projections
4.10.1 Kaczmarz's Method
4.10.2 Fixed Point of an Affine Transformation
4.11 Exercises
4.12 Projects
5 Least Squares and Matrix Geometry
5.1 The Column Space of a Matrix
5.2 Least Squares: Projection into Col(A)
5.3 Least Squares: Two Applications
5.3.1 Multiple Linear Regression
5.3.2 Curve Fitting with Least Squares
5.4 Four Fundamental Subspaces
5.4.1 Column-Row Factorization
5.5 Geometry of Transformations
5.6 Matrix Norms
5.7 Exercises
5.8 Project
6 Orthogonal Systems
6.1 Projections Revisited
6.2 Building Orthogonal Sets
6.3 QR Factorization
6.4 Least Squares with QR
6.5 Orthogonality and Matrix Norms
6.6 Exercises
6.7 Projects
7 Eigenvalues
7.1 Eigenvalues and Eigenvectors
7.2 Computing Eigenvalues
7.3 Computing Eigenvectors
7.4 Transformation of Eigenvalues
7.5 Eigenvalue Decomposition
7.6 Population Models
7.7 Rotations of R3
7.8 Existence of Eigenvalues
7.9 Exercises
8 Markov Processes
8.1 Stochastic Matrices
8.2 Stationary Distributions
8.3 The Power Method
8.4 Two-State Markov Processes
8.5 Ranking Web Pages
8.6 The Monte Carlo Method
8.7 Random Walks on Graphs
8.8 Exercises
8.9 Project
9 Symmetric Matrices
9.1 The Spectral Theorem
9.2 Norm of a Symmetric Matrix
9.3 Positive Semidefinite Matrices
9.3.1 Matrix Square Roots
9.4 Clusters in a Graph
9.5 Clustering a Graph with k-Means
9.6 Drawing a Graph