Linear Algebra for the AI Age
A Visual Journey Through the Mathematics of Intelligence
Master linear algebra through interactive visualizations. From vectors and matrices to eigenvalues, SVD, and the mathematics of attention mechanisms. Essential for AI, machine learning, computer vision, and beyond.
Geometric Foundations— Vectors, transformations, and matrices.
The Geometric Universe
What linear algebra really is - the big picture
Vectors - The Building Blocks
Vectors as geometric objects with operations that have meaning
Linear Transformations - The Big Idea
Transformations before matrices - the geometric foundation
Matrices - Recording Transformations
Matrices as transformation recorders, not just number grids
Linear Maps— Systems, spaces, rank, and inverse.
Systems of Equations - Geometry of Solutions
Systems as intersecting planes and lines
Vector Spaces - The Abstract Framework
Generalizing beyond arrows to abstract vector spaces
Linear Independence and Rank
The geometry of redundancy and dimension
The Inverse Matrix
Undoing transformations and when it is possible
Linear Maps Between Spaces
Abstraction and generalization of linear transformations
Decompositions— Eigenvalues, SVD, and factorizations.
Eigenvalues and Eigenvectors
The DNA of transformations - the most important concept for applications
Diagonalization
Simplifying transformations through eigenvector bases
Singular Value Decomposition
The universal factorization that works for any matrix
The Spectral Theorem
Why symmetric matrices are special
Matrix Decompositions
LU, QR, Cholesky and when to use each
Inner Products— Norms, orthogonality, projections, PCA.
Inner Products and Norms
Measuring angles and lengths in vector spaces
Orthogonality
Perpendicularity and its power in higher dimensions
Projections and Least Squares
The optimization workhorse of linear algebra
Principal Component Analysis
Dimensionality reduction through eigenanalysis
Advanced AI/ML— Tensors, matrix calculus, attention.
Tensors and Multilinear Algebra
Beyond matrices - the language of deep learning
Matrix Calculus
Gradients for optimization - the math behind backpropagation
The Mathematics of Attention
The linear algebra inside transformers
Numerical Linear Algebra
Making linear algebra work on computers
Applications— Computer vision, ML, and beyond.
Linear Algebra in Computer Vision
Images as matrices and beyond
Linear Algebra in Machine Learning
The mathematical core of ML algorithms
Linear Algebra Across Fields
Applications in graphs, signals, quantum, and more
134 sections. Begin with one.
Chapter 1 — The Geometric Universe — is where every reader starts.