Practical Linear Algebra for Data Science

Spis treści książki

• Preface
  • Conventions Used in This Book
  • Using Code Examples
  • OReilly Online Learning
  • How to Contact Us
  • Acknowledgments
• 1. Introduction
  • What Is Linear Algebra and Why Learn It?
  • About This Book
  • Prerequisites
    • Math
    • Attitude
    • Coding
  • Mathematical Proofs Versus Intuition from Coding
  • Code, Printed in the Book and Downloadable Online
  • Code Exercises
  • How to Use This Book (for Teachers and Self Learners)
• 2. Vectors, Part 1
  • Creating and Visualizing Vectors in NumPy
    • Geometry of Vectors
  • Operations on Vectors
    • Adding Two Vectors
    • Geometry of Vector Addition and Subtraction
    • Vector-Scalar Multiplication
    • Scalar-Vector Addition
      • The geometry of vector-scalar multiplication
    • Transpose
    • Vector Broadcasting in Python
  • Vector Magnitude and Unit Vectors
  • The Vector Dot Product
    • The Dot Product Is Distributive
    • Geometry of the Dot Product
  • Other Vector Multiplications
    • Hadamard Multiplication
    • Outer Product
    • Cross and Triple Products
  • Orthogonal Vector Decomposition
  • Summary
  • Code Exercises
• 3. Vectors, Part 2
  • Vector Sets
  • Linear Weighted Combination
  • Linear Independence
    • The Math of Linear Independence
    • Independence and the Zeros Vector
  • Subspace and Span
  • Basis
    • Definition of Basis
  • Summary
  • Code Exercises
• 4. Vector Applications
  • Correlation and Cosine Similarity
  • Time Series Filtering and Feature Detection
  • k-Means Clustering
  • Code Exercises
    • Correlation Exercises
    • Filtering and Feature Detection Exercises
    • k-Means Exercises
• 5. Matrices, Part 1
  • Creating and Visualizing Matrices in NumPy
    • Visualizing, Indexing, and Slicing Matrices
    • Special Matrices
  • Matrix Math: Addition, Scalar Multiplication, Hadamard Multiplication
    • Addition and Subtraction
    • Shifting a Matrix
    • Scalar and Hadamard Multiplications
  • Standard Matrix Multiplication
    • Rules for Matrix Multiplication Validity
    • Matrix Multiplication
    • Matrix-Vector Multiplication
      • Linear weighted combinations
      • Geometric transforms
  • Matrix Operations: Transpose
    • Dot and Outer Product Notation
  • Matrix Operations: LIVE EVIL (Order of Operations)
  • Symmetric Matrices
    • Creating Symmetric Matrices from Nonsymmetric Matrices
  • Summary
  • Code Exercises
• 6. Matrices, Part 2
  • Matrix Norms
    • Matrix Trace and Frobenius Norm
  • Matrix Spaces (Column, Row, Nulls)
    • Column Space
    • Row Space
    • Null Spaces
  • Rank
    • Ranks of Special Matrices
    • Rank of Added and Multiplied Matrices
    • Rank of Shifted Matrices
    • Theory and Practice
  • Rank Applications
    • In the Column Space?
    • Linear Independence of a Vector Set
  • Determinant
    • Computing the Determinant
    • Determinant with Linear Dependencies
    • The Characteristic Polynomial
  • Summary
  • Code Exercises
• 7. Matrix Applications
  • Multivariate Data Covariance Matrices
  • Geometric Transformations via Matrix-Vector Multiplication
  • Image Feature Detection
  • Summary
  • Code Exercises
    • Covariance and Correlation Matrices Exercises
    • Geometric Transformations Exercises
    • Image Feature Detection Exercises
• 8. Matrix Inverse
  • The Matrix Inverse
  • Types of Inverses and Conditions for Invertibility
  • Computing the Inverse
    • Inverse of a 2 × 2 Matrix
    • Inverse of a Diagonal Matrix
    • Inverting Any Square Full-Rank Matrix
    • One-Sided Inverses
  • The Inverse Is Unique
  • Moore-Penrose Pseudoinverse
  • Numerical Stability of the Inverse
  • Geometric Interpretation of the Inverse
  • Summary
  • Code Exercises
• 9. Orthogonal Matrices and QR Decomposition
  • Orthogonal Matrices
  • Gram-Schmidt
  • QR Decomposition
    • Sizes of Q and R
      • Why is upper triangular
    • QR and Inverses
  • Summary
  • Code Exercises
• 10. Row Reduction and LU Decomposition
  • Systems of Equations
    • Converting Equations into Matrices
    • Working with Matrix Equations
  • Row Reduction
    • Gaussian Elimination
    • Gauss-Jordan Elimination
    • Matrix Inverse via Gauss-Jordan Elimination
  • LU Decomposition
    • Row Swaps via Permutation Matrices
  • Summary
  • Code Exercises
• 11. General Linear Models and Least Squares
  • General Linear Models
    • Terminology
    • Setting Up a General Linear Model
  • Solving GLMs
    • Is the Solution Exact?
    • A Geometric Perspective on Least Squares
    • Why Does Least Squares Work?
  • GLM in a Simple Example
  • Least Squares via QR
  • Summary
  • Code Exercises
• 12. Least Squares Applications
  • Predicting Bike Rentals Based on Weather
    • Regression Table Using statsmodels
    • Multicollinearity
    • Regularization
  • Polynomial Regression
  • Grid Search to Find Model Parameters
  • Summary
  • Code Exercises
    • Bike Rental Exercises
    • Multicollinearity Exercise
    • Regularization Exercise
    • Polynomial Regression Exercise
    • Grid Search Exercises
• 13. Eigendecomposition
  • Interpretations of Eigenvalues and Eigenvectors
    • Geometry
    • Statistics (Principal Components Analysis)
    • Noise Reduction
    • Dimension Reduction (Data Compression)
  • Finding Eigenvalues
  • Finding Eigenvectors
    • Sign and Scale Indeterminacy of Eigenvectors
  • Diagonalizing a Square Matrix
  • The Special Awesomeness of Symmetric Matrices
    • Orthogonal Eigenvectors
    • Real-Valued Eigenvalues
  • Eigendecomposition of Singular Matrices
  • Quadratic Form, Definiteness, and Eigenvalues
    • The Quadratic Form of a Matrix
    • Definiteness
    • T Is Positive (Semi)definite
  • Generalized Eigendecomposition
  • Summary
  • Code Exercises
• 14. Singular Value Decomposition
  • The Big Picture of the SVD
    • Singular Values and Matrix Rank
  • SVD in Python
  • SVD and Rank-1 Layers of a Matrix
  • SVD from EIG
    • SVD of T
    • Converting Singular Values to Variance, Explained
    • Condition Number
  • SVD and the MP Pseudoinverse
  • Summary
  • Code Exercises
• 15. Eigendecomposition and SVD Applications
  • PCA Using Eigendecomposition and SVD
    • The Math of PCA
    • The Steps to Perform a PCA
    • PCA via SVD
  • Linear Discriminant Analysis
  • Low-Rank Approximations via SVD
    • SVD for Denoising
  • Summary
  • Exercises
    • PCA
    • Linear Discriminant Analyses
    • SVD for Low-Rank Approximations
    • SVD for Image Denoising
• 16. Python Tutorial
  • Why Python, and What Are the Alternatives?
  • IDEs (Interactive Development Environments)
  • Using Python Locally and Online
    • Working with Code Files in Google Colab
  • Variables
    • Data Types
    • Indexing
  • Functions
    • Methods as Functions
    • Writing Your Own Functions
    • Libraries
    • NumPy
    • Indexing and Slicing in NumPy
  • Visualization
  • Translating Formulas to Code
  • Print Formatting and F-Strings
  • Control Flow
    • Comparators
    • If Statements
      • elif and else
      • Multiple conditions
    • For Loops
    • Nested Control Statements
  • Measuring Computation Time
  • Getting Help and Learning More
    • What to Do When Things Go Awry
  • Summary
• Index

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