Linear Algebra Essentials for Machine Learning
Chapter 1: Vectors in Machine Learning
Vectors as Data Representations
Fundamental Vector Operations
Vector Magnitude and Direction
The Dot Product and Projections
Vector Norms: Measuring Length
Calculating Distances Between Vectors
Implementing Vector Operations with NumPy
Hands-on Practical: Feature Vector Manipulation
Quiz for Chapter 1
Chapter 2: Matrices: Data Representation and Transformations
Matrices for Organizing Data
Basic Matrix Operations
Matrix Multiplication Explained
Matrices as Linear Transformations
Common Types of Matrices
Representing Systems of Linear Equations
Implementing Matrix Operations with NumPy
Hands-on Practical: Transforming Data Points
Quiz for Chapter 2
Chapter 3: Solving Linear Systems and Matrix Inverses
Linear Systems in Machine Learning Models
Introduction to Gaussian Elimination
The Matrix Inverse
Calculating the Inverse of a Matrix
The Determinant and Invertibility
Solving Ax=b using the Inverse
Numerical Stability and Alternatives
Hands-on Practical: Solving for Model Coefficients
Quiz for Chapter 3
Chapter 4: Vector Spaces, Subspaces, and Linear Independence
Defining Vector Spaces
Understanding Subspaces
Linear Combinations and Span
Linear Independence of Vectors
Basis and Dimension
Column Space and Null Space
Rank of a Matrix
Hands-on Practical: Analyzing Feature Vector Sets
Quiz for Chapter 4
Chapter 5: Eigenvalues and Eigenvectors
Definition of Eigenvalues and Eigenvectors
Geometric Interpretation
The Characteristic Equation
Calculating Eigenvectors
Eigen-decomposition of a Matrix
Significance in Principal Component Analysis (PCA)
Calculating Eigenvalues/Eigenvectors with NumPy
Hands-on Practical: Eigen-decomposition Calculations
Quiz for Chapter 5
Chapter 6: Matrix Decompositions for Machine Learning
Introduction to Matrix Decompositions
Singular Value Decomposition (SVD)
Geometric Interpretation of SVD
SVD for Dimensionality Reduction
SVD Applications in Data Compression
Relationship between SVD and Eigen-decomposition
Overview of LU Decomposition
Overview of QR Decomposition
Implementing Decompositions with SciPy/NumPy
Hands-on Practical: Applying SVD
Quiz for Chapter 6
Significance in Principal Component Analysis (PCA)
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- On Lines and Planes of Closest Fit to Systems of Points in Space, Karl Pearson, 1901 Philosophical Magazine, Series 6, Vol. 2 (Taylor & Francis) DOI: 10.1080/14786440109462720 - A foundational paper that geometrically introduced the concept of principal axes, which is the precursor to Principal Component Analysis.
- The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Trevor Hastie, Robert Tibshirani, and Jerome Friedman, 2009 (Springer) - A widely respected textbook in statistical learning, providing a thorough treatment of PCA as a dimensionality reduction technique within a broader machine learning context.
- Linear Algebra and Learning from Data, Gilbert Strang, 2019 (Wellesley-Cambridge Press) - A textbook directly connecting linear algebra concepts like eigenvalues and eigenvectors to their use in machine learning algorithms, including PCA.
- Principal Component Analysis (PCA), Andrew Ng, 2020 Stanford CS229 Machine Learning Course Notes (Stanford University) - Course notes from a highly influential machine learning curriculum, presenting a practical and intuitive explanation of PCA's mathematical foundations and applications.