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An edition of Coding the Matrix (2013)

Coding the Matrix

Linear Algebra through Computer Science Applications

by Philip N. Klein

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An edition of Coding the Matrix (2013)

Coding the Matrix

Linear Algebra through Computer Science Applications

by Philip N. Klein

★★★★★ 5.0 (1 rating)
4 Want to read
1 Currently reading
1 Have read

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Publish Date

Jul 26, 2013

Publisher

Newtonian Press

Pages

528

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1 Coding the Matrix: Linear Algebra through Applications to Computer Science Sep 03, 2013, Newtonian Press paperback 0615880991 9780615880990	zzzz Locate
2 Coding the Matrix: Linear Algebra through Computer Science Applications Jul 26, 2013, Newtonian Press paperback 061585673X 9780615856735	aaaa Borrow Listen Locate

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Table of Contents

0. The Function (and Other Mathematical and Computational Preliminaries)

0.1. Set Terminology and Notation

0.2. Cartesian Product

0.3. The Function

0.3.1. Functions Versus Procedures, Versus Computational Problems

0.3.2. The Two Computational Problems Related to a Function

0.3.3. Notation for the Set of Functions with Given Domain and Co-domain

0.3.4. Identity Function

0.3.5. Composition of Functions

0.3.6. Associativity of Function Composition

0.3.7. Functional Inverse

0.4. Probability

0.4.1. Probability Distributions

0.4.2. Events, and Adding Probabilities

0.4.3. Applying a Function to a Random Input

0.4.4. Perfect Secrecy

0.4.5. Perfect Secrecy and Invertible Functions

0.5. Lab: Introduction to Python—Sets, Lists, Dictionaries, and Comprehensions

0.5.1. Simple Expressions

0.5.2. Assignment Statements

0.5.3. Conditional Expressions

0.5.7. Other Things to Iterate Over

0.5.8. Dictionaries

0.5.9. Defining One-Line Procedures

0.6. Lab: Python Modules and Control Structures — and Inverse Index

0.6.1. Using Existing Modules

0.6.2. Creating Your Own Modules

0.6.3. Loops and Conditional Statements

0.6.4. Grouping in Python Using Indentation

0.6.5. Breaking Out of a Loop

0.6.6. Reading from a File

0.6.7. Mini-Search Engine

0.7. Review Questions

1.1. Introduction to Complex Numbers

1.2. Complex Numbers in Python

1.3. Abstracting over Fields

1.4. Playing with C

1.4.1. The Absolute Value of a Complex Number

1.4.2. Adding Complex Numbers

1.4.3. Multiplying Complex Numbers by a Positive Real Number

1.4.4. Multiplying Complex Numbers by a Negative Number: Rotation by 180 Degrees

1.4.5. Multiplying by i: Rotation by 90 Degrees

1.4.6. The Unit Circle in the Complex Plane: Argument and Angle

1.4.7. Euler's Formula

1.4.8. Polar Representation for Complex Numbers

1.4.9. The First Law of Exponentiation

1.4.10. Rotation by θ Radians

1.4.11. Combining Operations

1.4.12. Beyond Two Dimensions

1.5. Playing with GF(2)

1.5.1. Perfect Secrecy Revisited

1.5.2. Network Coding

1.6. Review Questions

2.1. What Is a Vector?

2.2. Vectors Are Functions

2.2.1. Representation of Vectors Using Python Dictionaries

2.2.2. Sparsity

2.3. What Can We Represent with Vectors?

2.4. Vector Addition

2.4.1. Translation and Vector Addition

2.4.2. Vector Addition Is Associative and Commutative

2.4.3. Vectors as Arrows

2.5. Scalar-Vector Multiplication

2.5.1. Scaling Arrows

2.5.2. Associativity of Scalar-Vector Multiplication

2.5.3. Line Segments through the Origin

2.5.4. Lines through the Origin

2.6. Combining Vector Addition and Scalar Multiplication

2.6.1. Line Segments and Lines That Don't Go through the Origin

2.6.2. Distributive Laws for Scalar-Vector Multiplication and Vector Addition

2.6.3. First Look at Convex Combinations

2.6.4. First Look at Affine Combinations

2.7. Dictionary-Based Representations of Vectors

2.7.1. Setter and Getter

2.7.2. Scalar-Vector Multiplication

2.7.3. Addition

2.7.4. Vector Negative, Invertibility of Vector Addition, and Vector Subtraction

2.8. Vectors over GF(2)

2.8.1. Perfect Secrecy Re-Revisited

2.8.2. All-or-Nothing Secret-Sharing Using GF(2)

2.8.3. Lights Out

2.9. Dot-Product

2.9.1. Total Cost or Benefit

2.9.2. Linear Equations

2.9.3. Measuring Similarity

2.9.4. Dot-Product of Vectors over GF(2)

2.9.5. Parity Bit

2.9.6. Simple Authentication Scheme

2.9.7. Attacking the Simple Authentication Scheme

2.9.8. Algebraic Properties of the Dot-Product

2.9.9. Attacking the Simple Authentication Scheme, Revisited

2.10. Our Implementation of Vec

2.10.1. Syntax for Manipulating Vecs

2.10.2. The Implementation

2.10.3. Using Vecs

2.10.4. Printing Vecs

2.10.5. Copying Vecs

2.10.6. From List to Vec

2.11. Solving a Triangular System of Linear Equations

2.11.1. Upper-Triangular Systems

2.11.2. Backward Substitution

2.11.3. First Implementation of Backward Substitution

2.11.4. When Does the Algorithm Work?

2.11.5. Backward Substitution with Arbitrary-Domain Vectors

2.12. Lab: Comparing Voting Records Using Dot-Product

2.12.1. Motivation

2.12.2. Reading in the File

2.12.3. Two Ways to Use Dot-Product to Compare Vectors

2.12.4. Policy Comparison

2.12.5. Not Your Average Democrat

2.12.6. Bitter Rivals

2.12.7. Open-Ended Study

2.13. Review Questions

3. The Vector Space

3.1. Linear Combination

3.1.1. Definition of Linear Combination

3.1.2. Uses of Linear Combinations

3.1.3. From Coefficients to Linear Combination

3.1.4. From Linear Combination to Coefficients

3.2.1. Definition of Span

3.2.2. A System of Linear Equations Implies Other Equations

3.2.3. Generators

3.2.4. Linear Combinations of Linear Combinations

3.2.5. Standard Generators

3.3. The Geometry of Sets of Vectors

3.3.1. The Geometry of the Span of Vectors over R

3.3.2. The Geometry of Solution Sets of Homogeneous Linear Systems

3.3.3. The Two Representations of Flats Containing the Origin

3.4. Vector Spaces

3.4.1. What's Common to the Two Representations?

3.4.2. Definition and Examples of Vector Space

3.4.3. Subspaces

3.4.4. Abstract Vector Spaces

3.5. Affine Spaces

3.5.1. Flats That Don't Go through the Origin

3.5.2. Affine Combinations

3.5.3. Affine Spaces

3.5.4. Representing an Affine Space as the Solution Set of a Linear System

3.5.5. The Two Representations, Revisited

3.6. Linear Systems, Homogeneous and Otherwise

3.6.1. The Homogeneous Linear System Corresponding to a General Linear System

3.6.2. Number of Solutions Revisited

3.6.3. Towards Intersecting a Plane and a Line

3.6.4. Checksum Functions

3.7. Review Questions

4.1. What Is a Matrix?

4.1.1. Traditional Matrices

4.1.2. The Matrix Revealed

4.1.3. Rows, Columns, and Entries

4.1.4. Our Python Implementation of Matrices

4.1.5. Identity Matrix

4.1.6. Converting between Matrix Representations

4.1.7. `matutil.py`

4.2. Column Space and Row Space

4.3. Matrices as Vectors

4.5. Matrix-Vector and Vector-Matrix Multiplication in Terms of Linear Combinations

4.5.1. Matrix-Vector Multiplication in Terms of Linear Combinations

4.5.2. Vector-Matrix Multiplication in Terms of Linear Combinations

4.5.3. Formulating Expressing a Given Vector as a Linear Combination as a Matrix-Vector Equation

4.5.4. Solving a Matrix-Vector Equation

4.6. Matrix-Vector Multiplication in Terms of Dot-Products

4.6.1. Definitions

4.6.2. Example Applications

4.6.3. Formulating a System of Linear Equations as a Matrix-Vector Equation

4.6.4. Triangular Systems and Triangular Matrices

4.6.5. Algebraic Properties of Matrix-Vector Multiplication

4.7. Null Space

4.7.1. Homogeneous Linear Systems and Matrix Equations

4.7.2. The Solution Space of a Matrix-Vector Equation

4.7.3. Introduction to Error-Correcting Codes

4.7.4. Linear Codes

4.7.5. The Hamming Code

4.8. Computing Sparse Matrix-Vector Product

4.9. The Matrix Meets the Function

4.9.1. From Matrix to Function

4.9.2. From Function to Matrix

4.9.3. Examples of Deriving the Matrix

4.10. Linear Functions

4.10.1. Which Functions Can Be Expressed as a Matrix-Vector Product

4.10.2. Definition and Simple Examples

4.10.3. Linear Functions and Zero Vectors

4.10.4. What Do Linear Functions Have to Do with Lines?

4.10.5. Linear Functions That Are One-to-One

4.10.6. Linear Functions That Are Onto

4.10.7. A Linear Function from $F^C$ to $F^R$ Can Be Represented by a Matrix

4.10.8. Diagonal Matrices

4.11. Matrix-Matrix Multiplication

4.11.1. Matrix-Matrix Multiplication in Terms of Matrix-Vector and Vector-Matrix Multiplication

4.11.2. Matrix-Matrix Multiplication and Function Composition

4.11.3. Transpose of Matrix-Matrix Product

4.11.4. Column Vector and Row Vector

4.11.5. Every Vector Is Interpreted as a Column Vector

4.11.6. Linear Combinations of Linear Combinations Revisited

4.12. Inner Product and Outer Product

4.12.1. Inner Product

4.12.2. Outer Product

4.13. From Function Inverse to Matrix Inverse

4.13.1. The Inverse of a Linear Function Is Linear

4.13.2. The Matrix Inverse

4.13.3. Uses of Matrix Inverse

4.13.4. More about Matrix Inverse

4.14. Lab: Error-Correcting Codes

4.14.1. The Check Matrix

4.14.2. The Generator Matrix

4.14.3. Hamming's Code

4.14.4. Decoding

4.14.5. Error Syndrome

4.14.6. Finding the Error

4.14.7. Putting It All Together

4.15. Lab: Transformations in 2D Geometry

4.15.1. Our Representation for Points in the Plane

4.15.2. Transformations

4.15.3. Image Representation

4.15.4. Loading and Displaying Images

4.15.5. Linear Transformations

4.15.6. Translation

4.15.7. Scaling

4.15.8. Rotation

4.15.9. Rotation about a Center Other than the Origin

4.15.10. Reflection

4.15.11. Color Transformations

4.15.12. Reflection More Generally

4.16. Review Questions

5.1. Coordinate Systems

5.1.1. René Descartes' Idea

5.1.2. Coordinate Representation

5.1.3. Coordinate Representation and Matrix-Vector Multiplication

5.2. First Look at Lossy Compression

5.2.1. Strategy 1: Replace Vector with Closest Sparse Vector

5.2.2. Strategy 2: Represent Image Vector by Its Coordinate Representation

5.3. Two Greedy Algorithms for Finding a Set of Generators

5.3.1. Grow Algorithm

5.3.2. Shrink Algorithm

5.3.3. When Greed Fails

5.4. Minimum Spanning Forest and GF(2)

5.4.1. Definitions

5.4.2. The Grow Algorithm and the Shrink Algorithm for Minimum Spanning Forest

5.4.3. Formulating Minimum Spanning Forest in Linear Algebra

5.5. Linear Dependence

5.5.1. The Superfluous-Vector Lemma

5.5.2. Defining Linear Dependence

5.5.3. Linear Dependence in Minimum Spanning Forest

5.5.4. Properties of Linear (In)dependence

5.5.5. Analyzing the Grow Algorithm

5.5.6. Analyzing the Shrink Algorithm

5.6.1. Defining Basis

5.6.2. The Standard Basis for F^L

5.6.3. Towards Showing That Every Vector Space Has a Basis

5.6.4. Any Finite Set of Vectors Contains a Basis for Its Span

5.6.5. Any Linearly Independent Subset of Vectors Belonging to V Can Be Extended to Form a Basis for V

5.7. Unique Representation

5.7.1. Uniqueness of Representation in Terms of a Basis

5.8. Change of Basis, First Look

5.8.1. The Function from Representation to Vector

5.8.2. From One Representation to Another

5.9. Perspective Rendering

5.9.1. Points in the World

5.9.2. The Camera and the Image Plane

5.9.3. The Camera Coordinate System

5.9.4. From the Camera Coordinates of a Point in the Scene to the Camera Coordinates of the Corresponding Point in the Image Plane

5.9.5. From World Coordinates to Camera Coordinates

5.9.6. To Pixel Coordinates

5.10. Computational Problems Involving Finding a Basis

5.11. The Exchange Lemma

5.11.1. The Lemma

5.11.2. Proof of Correctness of the Grow Algorithm for MSF

5.12. Lab: Perspective Rectification

5.12.1. The Camera Basis

5.12.2. The Whiteboard Basis

5.12.3. Mapping from Pixels to Points on the Whiteboard

5.12.4. Mapping a Point Not on the Whiteboard to the Corresponding Point on the Whiteboard

5.12.5. The Change-of-Basis Matrix

5.12.6. Computing the Change-of-Basis Matrix

5.12.7. Image Representation

5.12.8. Synthesizing the Perspective-Free Image

5.13. Review Questions

6.1. The Size of a Basis

6.1.1. The Morphing Lemma and Its Implications

6.1.2. Proof of the Morphing Lemma

6.2. Dimension and Rank

6.2.1. Definitions and Examples

6.2.2. Geometry

6.2.3. Dimension and Rank in Graphs

6.2.4. The Cardinality of a Vector Space over GF(2)

6.2.5. The Dimension Principle

6.2.6. For a Finite Set D, Every Subspace of F^D Has a Basis

6.2.7. The Rank Theorem

6.2.8. Simple Authentication Revisited

6.3. Direct Sum

6.3.1. Definition

6.3.2. Generators for the Direct Sum

6.3.3. Basis for the Direct Sum

6.3.4. Unique Decomposition of a Vector

6.3.5. Complementary Subspaces

6.4. Dimension and Linear Functions

6.4.1. Linear Function Invertibility

6.4.2. The Largest Invertible Subfunction

6.4.3. The Kernel-Image Theorem

6.4.4. Linear Function Invertibility, Revisited

6.4.5. The Rank-Nullity Theorem

6.4.6. Checksum Problem Revisited

6.4.7. Matrix Invertibility

6.4.8. Matrix Invertibility and Change of Basis

6.5.1. Conversions between Representations

6.5.2. The Dual of a Vector Space

6.5.3. The Dual Dimension Theorem

6.5.4. From Generators for V to Generators for V*, and Vice Versa

6.5.5. The Duality Theorem

6.6. Review Questions

7. Gaussian Elimination

7.1. Echelon Form

7.1.1. From Echelon Form to a Basis for Row Space

7.1.2. Rowlist in Echelon Form

7.1.3. Sorting Rows by Position of the Leftmost Nonzero

7.1.4. Elementary Row-Addition Operations

7.1.5. Multiplying by an Elementary Row-Addition Matrix

7.1.6. Row-Addition Operations Preserve Row Space

7.1.7. Basis, Rank, and Linear Independence through Gaussian Elimination

7.1.8. When Gaussian Elimination Fails

7.1.9. Pivoting, and Numerical Analysis

7.2. Gaussian Elimination over GF(2)

7.3. Using Gaussian Elimination for Other Problems

7.3.1. There Is an Invertible Matrix M Such That M A Is in Echelon Form

7.3.2. Computing M without Matrix Multiplications

7.4. Solving a Matrix-Vector Equation Using Gaussian Elimination

7.4.1. Solving a Matrix-Vector Equation When the Matrix Is in Echelon Form: The Invertible Case

7.4.2. Coping with Zero Rows

7.4.3. Coping with Irrelevant Columns

7.4.4. Finding a Basis for the Null Space

7.5. Factoring Integers

7.5.1. First Attempt at Factoring

7.6. Lab: Threshold Secret-Sharing

7.6.1. First Attempt

7.6.2. Scheme That Works

7.6.3. Implementing the Scheme

7.6.4. Generating u

7.6.5. Finding Vectors That Satisfy the Requirement

7.6.6. Sharing a String

7.7. Lab: Factoring Integers

7.7.1. First Attempt to Use Square Roots

7.7.2. Euclid's Algorithm for Greatest Common Divisor

7.7.3. Using Square Roots Revisited

7.8. Review Questions

8. The Inner Product

8.1. The Fire Engine Problem

8.1.1. Distance, Length, Norm, Inner Product

8.2. The Inner Product for Vectors over the Reals

8.2.1. Norms of Vectors over the Reals

8.3. Orthogonality

8.3.1. Properties of Orthogonality

8.3.2. Decomposition of b into Parallel and Perpendicular Components

8.3.3. Orthogonality Property of the Solution to the Fire Engine Problem

8.3.4. Finding the Projection and the Closest Point

8.3.5. Solution to the Fire Engine Problem

8.3.6. Outer Product and Projection

8.3.7. Towards Solving the Higher-Dimensional Version

8.4. Lab: Machine Learning

8.4.1. The Data

8.4.2. Supervised Learning

8.4.3. Selecting the Classifier That Minimizes the Error on the Training Data

8.4.4. Nonlinear Optimization by Hill-Climbing

8.4.5. Gradient

8.4.6. Getting the Data

8.4.7. Evaluating a Hypothesis

8.4.8. Finding a Good Hypothesis

8.5. Review Questions

9. Orthogonalization

9.1. Projection Orthogonal to Multiple Vectors

9.1.1. Orthogonal to a Set of Vectors

9.1.2. Projecting onto and Orthogonal to a Vector Space

9.1.3. First Attempt at Projecting Orthogonal to a List of Vectors

9.2. Projecting b Orthogonal to a List of Mutually Orthogonal Vectors

9.2.1. Proving the Correctness of project_orthogonal

9.2.2. Augmenting project_orthogonal

9.3. Building an Orthogonal Set of Generators

9.3.1. The orthogonalize Procedure

9.3.2. Proving the Correctness of orthogonalize

9.4. Solving the Computational Problem: Closest Point in the Span of Many Vectors

9.5. Solving Other Problems Using orthogonalize

9.5.1. Computing a Basis

9.5.2. Computing a Subset Basis

9.5.3. augmented_orthogonalize

9.5.4. Algorithms That Work in the Presence of Rounding Errors

9.6. Orthogonal Complement

9.6.1. Definition of Orthogonal Complement

9.6.2. Orthogonal Complement and Direct Sum

9.6.3. Normal to a Plane in R^3 Given as Span or Affine Hull

9.6.4. Orthogonal Complement and Null Space and Dual Space

9.6.5. Normal to a Plane in R^3 Given by an Equation

9.6.6. Computing the Orthogonal Complement

9.7. The QR Factorization

9.7.1. Orthogonal and Column-Orthogonal Matrices

9.7.2. Defining the QR Factorization of a Matrix

9.7.3. Requiring A to Have Linearly Independent Columns

9.8. Using the QR Factorization to Solve a Matrix Equation Ax = b

9.8.1. The Square Case

9.8.2. Correctness in the Square Case

9.8.3. The Least-Squares Problem

9.8.4. The Coordinate Representation in Terms of the Columns of a Column-Orthogonal Matrix

9.8.5. Using QR_solve When A Has More Rows Than Columns

9.9. Applications of Least Squares

9.9.1. Linear Regression (Line-Fitting)

9.9.2. Fitting to a Quadratic

9.9.3. Fitting to a Quadratic in Two Variables

9.9.4. Coping with Approximate Data in the Industrial Espionage Problem

9.9.5. Coping with Approximate Data in the Sensor Node Problem

9.9.6. Using the Method of Least Squares in the Machine-Learning Problem

9.10. Review Questions

10. Special Bases

10.1. Closest k-Sparse Vector

10.2. Closest Vector Whose Representation with Respect to a Given Basis Is k-Sparse

10.2.1. Finding the Coordinate Representation in Terms of an Orthonormal Basis

10.2.2. Multiplication by a Column-Orthogonal Matrix Preserves Norm

10.3.1. One-Dimensional "Images" of Different Resolutions

10.3.2. Decomposing V_n as a Direct Sum

10.3.3. The Wavelet Bases

10.3.4. The Basis for V_1

10.3.5. General n

10.3.6. The First Stage of Wavelet Transformation

10.3.7. The Subsequent Levels of Wavelet Decomposition

10.3.8. Normalizing

10.3.9. The Backward Transform

10.3.10. Implementation

10.4. Polynomial Evaluation and Interpolation

10.5. Fourier Transform

10.6. Discrete Fourier Transform

10.6.1. The Laws of Exponentiation

10.6.2. The n Stopwatches

10.6.3. Discrete Fourier Space: Sampling the Basis Functions

10.6.4. The Inverse of the Fourier Matrix

10.6.5. The Fast Fourier Algorithm

10.7. The Inner Product for the Field of Complex Numbers

10.8. Circulant Matrices

10.8.1. Multiplying a Circulant Matrix by a Column of the Fourier Matrix

10.8.2. Circulant Matrices and Change of Basis

10.9. Lab: Using Wavelets for Compression

10.9.1. Unnormalized Forward Transform

10.9.2. Normalization in the Forward Transform

10.9.3. Compression by Suppression

10.9.4. Unnormalizing

10.9.5. Unnormalized Backward Transform

10.9.6. Backward Transform

10.9.7. Auxiliary Procedure

10.9.8. Two-Dimensional Wavelet Transform

10.9.9. Forward Two-Dimensional Transform

10.9.10. More Auxiliary Procedures

10.9.11. Two-Dimensional Backward Transform

10.9.12. Experimenting with Compression of Images

10.10. Review Questions

10.11. Problems

11. The Singular Value Decomposition

11.1. Approximation of a Matrix by a Low-Rank Matrix

11.1.1. The Benefits of Low-Rank Matrices

11.1.2. Matrix Norm

11.2. The Trolley-Line-Location Problem

11.2.1. Solution to the Trolley-Line-Location Problem

11.2.2. Rank-One Approximation to a Matrix

11.2.3. The Best Rank-One Approximation

11.2.4. An Expression for the Best Rank-One Approximation

11.2.5. The Closest One-Dimensional Affine Space

11.3. Closest Dimension-k Vector Space

11.3.1. A Gedanken Algorithm to Find the Singular Values and Vectors

11.3.2. Properties of the Singular Values and Right Singular Vectors

11.3.3. The Singular Value Decomposition

11.3.4. Using Right Singular Vectors to Find the Closest k-Dimensional Space

11.3.5. Best Rank-k Approximation to A

11.3.6. Matrix Form for Best Rank-k Approximation

11.3.7. Number of Nonzero Singular Values Is Rank A

11.3.8. Numerical Rank

11.3.9. Closest k-Dimensional Affine Space

11.3.10. Proof That U Is Column-Orthogonal

11.4. Using the Singular Value Decomposition

11.4.1. Using SVD to Do Least Squares

11.6. Lab: Eigenfaces

11.7. Review Questions

12. The Eigenvector

12.1. Modeling Discrete Dynamic Processes

12.1.1. Two Interest-Bearing Accounts

12.1.2. Fibonacci Numbers

12.2. Diagonalization of the Fibonacci Matrix

12.3. Eigenvalues and Eigenvectors

12.3.1. Similarity and Diagonalizability

12.4. Coordinate Representation in Terms of Eigenvectors

12.5. The Internet Worm

12.6. Existence of Eigenvalues

12.6.1. Positive-Definite Matrices

12.6.2. Matrices with Distinct Eigenvalues

12.6.3. Symmetric Matrices

12.6.4. Upper-Triangular Matrices

12.6.5. General Square Matrices

12.7. Power Method

12.8. Markov Chains

12.8.1. Modeling Population Movement

12.8.2. Modeling Randy

12.8.3. Markov Chain Definitions

12.8.4. Modeling Spatial Locality in Memory Fetches

12.8.5. Modeling Documents: Hamlet in Wonderland

12.8.6. Modeling Lots of Other Stuff

12.8.7. Stationary Distributions of Markov Chains

12.8.8. Sufficient Condition for Existence of a Stationary Distribution

12.9. Modeling a Web Surfer: PageRank

12.10. *The Determinant

12.10.1. Areas of Parallelograms

12.10.2. Volumes of Parallelepipeds

12.10.3. Expressing the Area of a Polygon in Terms of Areas of Parallelograms

12.10.4. The Determinant

12.10.5. *Characterizing Eigenvalues via the Determinant Function

12.11. *Proofs of Some Eigentheorems

12.11.1. Existence of Eigenvalues

12.11.2. Diagonalization of Symmetric Matrices

12.11.3. Triangularization

12.12. Lab: PageRank

12.12.1. Concepts

12.12.2. Working with a Big Dataset

12.12.3. Implementing PageRank Using the Power Method

12.12.4. The Dataset

12.12.5. Handling Queries

12.12.6. Biasing the PageRank

12.12.7. Optional: Handling Multiword Queries

12.13. Review Questions

12.14. Problems

13. The Linear Program

13.1. The Diet Problem

13.2. Formulating the Diet Problem as a Linear Program

13.3. The Origins of Linear Programming

13.3.1. Terminology

13.3.2. Linear Programming in Different Forms

13.3.3. Integer Linear Programming

13.4. Geometry of Linear Programming: Polyhedra and Vertices

13.5. There Is an Optimal Solution That Is a Vertex of the Polyhedron

13.6. An Enumerative Algorithm for Linear Programming

13.7. Introduction to Linear-Programming Duality

13.8. The Simplex Algorithm

13.8.1. Termination

13.8.2. Representing the Current Solution

13.8.3. A Pivot Step

13.8.4. Simple Example

13.9. Finding a Vertex

13.10. Game Theory

13.11. Formulation as a Linear Program

13.12. Nonzero-Sum Games

13.13. Lab: Learning through Linear Programming

13.13.1. Reading in the Training Data

13.13.2. Setting up the Linear Program

13.13.3. Main Constraints

13.13.4. Nonnegativity Constraints

13.13.5. The Matrix A

13.13.6. The Right-Hand Side Vector b

13.13.7. The Objective Function Vector c

13.13.8. Putting It Together

13.13.9. Finding a Vertex

13.13.10. Solving the Linear Program

13.13.11. Using the Result

13.14. Compressed Sensing

13.14.1. More Quickly Acquiring an MRI Image

13.14.2. Computation to the Rescue

13.14.3. Onwards

13.15. Review Questions

13.16. Problems

Edition Notes

Source title: Coding the Matrix: Linear Algebra through Computer Science Applications

The Physical Object

Format: paperback
Number of pages: 528

Edition Identifiers

Open Library: OL30400578M
Internet Archive: codingmatrixline0000phil
ISBN 10: 061585673X
ISBN 13: 9780615856735

Work Identifiers

Work ID: OL22321354W

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Created September 21, 2020
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