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Book · Beginner · 60+ hours

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.

25Chapters
134Sections
37hReading
6Parts
Start chapter 01Browse curriculum
Part IGeometric Foundations23Part IILinear Maps24Part IIIDecompositions26Part IVInner Products22Part VAdvanced AI/ML21Part VIApplications18
Curriculum

25 chapters— in publication order.

Part I

Geometric Foundations

Vectors, transformations, and matrices

  1. 01The Geometric Universe5 sections · 65m

    What linear algebra really is - the big picture

    1.1–1.5
    1.1What Linear Algebra Really Is1.2Vectors as Arrows vs Vectors as Lists1.3The Power of Linearity1.4A Tour of Applications1.5How to Use This Book
  2. 02Vectors - The Building Blocks6 sections · 82m

    Vectors as geometric objects with operations that have meaning

    2.1–2.6
    2.1Vectors in 2D and 3D2.2Vector Addition: The Parallelogram Rule2.3Scalar Multiplication: Stretching and Flipping2.4Linear Combinations: The Heart of LA2.5Basis Vectors: The Coordinate System2.6Higher Dimensions: Beyond What We See
  3. 03Linear Transformations - The Big Idea6 sections · 90m

    Transformations before matrices - the geometric foundation

    3.1–3.6
    3.1What is a Linear Transformation?3.2Transformations in 2D3.3Rotation, Scaling, Shearing3.4Composition of Transformations3.5The Inverse Transformation3.63D Transformations
  4. 04Matrices - Recording Transformations6 sections · 95m

    Matrices as transformation recorders, not just number grids

    4.1–4.6
    4.1The Matrix as a Transformation Record4.2Reading Transformations from Matrices4.3Matrix-Vector Multiplication4.4Matrix-Matrix Multiplication4.5Special Matrices: Identity, Zero, Diagonal4.6The Determinant as Area/Volume Change

Part II

Linear Maps

Systems, spaces, rank, and inverse

  1. 05Systems of Equations - Geometry of Solutions5 sections · 86m

    Systems as intersecting planes and lines

    5.1–5.5
    5.1Equations as Geometric Constraints5.2Row Picture vs Column Picture5.3Gaussian Elimination Geometrically5.4When Solutions Exist: Rank and Nullspace5.5Overdetermined and Underdetermined Systems
  2. 06Vector Spaces - The Abstract Framework5 sections · 86m

    Generalizing beyond arrows to abstract vector spaces

    6.1–6.5
    6.1Beyond Arrows: Functions as Vectors6.2The Eight Axioms Made Intuitive6.3Subspaces: Spaces Within Spaces6.4Span, Independence, Basis, Dimension6.5Coordinates and Change of Basis
  3. 08Linear Independence and Rank5 sections · 72m

    The geometry of redundancy and dimension

    8.1–8.5
    8.1When Vectors Are Redundant8.2Linear Independence Geometrically8.3Rank: Dimension of the Image8.4The Rank-Nullity Theorem8.5Computing Rank in Practice
  4. 09The Inverse Matrix5 sections · 75m

    Undoing transformations and when it is possible

    9.1–9.5
    9.1When Can We Undo?9.2Computing the Inverse9.3Properties of the Inverse9.4The Determinant and Invertibility9.5Numerical Stability and Condition Numbers
  5. 10Linear Maps Between Spaces4 sections · 66m

    Abstraction and generalization of linear transformations

    10.1Domain, Codomain, Image, Kernel10.2Injective, Surjective, Bijective10.3Isomorphisms: When Spaces Are The Same10.4Matrix Representation of Linear Maps

Part III

Decompositions

Eigenvalues, SVD, and factorizations

  1. 11Eigenvalues and Eigenvectors6 sections · 101m

    The DNA of transformations - the most important concept for applications

    11.1–11.6
    11.1The Eigenvector Intuition11.2Finding Eigenvalues: The Characteristic Polynomial11.3Finding Eigenvectors11.4Geometric Meaning of Eigenvalues11.5Complex Eigenvalues: Rotation in Disguise11.6Eigenspaces and Multiplicity
  2. 12Diagonalization5 sections · 80m

    Simplifying transformations through eigenvector bases

    12.1–12.5
    12.1The Dream: Diagonal Matrices12.2The Diagonalization Process12.3When Diagonalization Fails12.4Powers of Matrices Made Easy12.5Applications: Fibonacci, PageRank, Markov
  3. 13Singular Value Decomposition6 sections · 106m

    The universal factorization that works for any matrix

    13.1–13.6
    13.1What SVD Tells Us13.2Computing the SVD13.3Singular Values and Matrix Properties13.4Low-Rank Approximation13.5Image Compression with SVD13.6The Pseudoinverse
  4. 14The Spectral Theorem5 sections · 81m

    Why symmetric matrices are special

    14.1–14.5
    14.1Symmetric Matrices: Real Eigenvalues14.2Orthogonal Eigenvectors14.3The Spectral Decomposition14.4Quadratic Forms14.5Positive Definite Matrices
  5. 15Matrix Decompositions4 sections · 63m

    LU, QR, Cholesky and when to use each

    15.1LU Decomposition15.2QR Decomposition15.3Cholesky Decomposition15.4Choosing the Right Decomposition

Part IV

Inner Products

Norms, orthogonality, projections, PCA

  1. 16Inner Products and Norms5 sections · 75m

    Measuring angles and lengths in vector spaces

    16.1–16.5
    16.1The Dot Product: Measuring Alignment16.2General Inner Products16.3Norms: Measuring Size16.4The Cauchy-Schwarz Inequality16.5Angles Between Vectors
  2. 17Orthogonality5 sections · 77m

    Perpendicularity and its power in higher dimensions

    17.1–17.5
    17.1Orthogonal Vectors17.2Orthogonal and Orthonormal Bases17.3Orthogonal Matrices17.4The Gram-Schmidt Process17.5Orthogonal Complements
  3. 18Projections and Least Squares6 sections · 104m

    The optimization workhorse of linear algebra

    18.1–18.6
    18.1Projecting onto a Line18.2Projecting onto a Subspace18.3The Projection Matrix18.4Least Squares: When Ax=b Has No Solution18.5Linear Regression as Projection18.6Regularization: Ridge and LASSO
  4. 19Principal Component Analysis6 sections · 95m

    Dimensionality reduction through eigenanalysis

    19.1–19.6
    19.1The Goal: Finding Principal Directions19.2Covariance and the Covariance Matrix19.3PCA Algorithm Step-by-Step19.4Choosing the Number of Components19.5PCA for Visualization19.6Limitations of PCA

Part V

Advanced AI/ML

Tensors, matrix calculus, attention

  1. 20Tensors and Multilinear Algebra5 sections · 91m

    Beyond matrices - the language of deep learning

    20.1–20.5
    20.1What is a Tensor?20.2Tensor Operations20.3Tensor Decompositions20.4Tensors in Deep Learning20.5Einstein Notation
  2. 21Matrix Calculus5 sections · 89m

    Gradients for optimization - the math behind backpropagation

    21.1–21.5
    21.1Derivatives of Scalar Functions of Matrices21.2The Gradient and the Jacobian21.3The Chain Rule for Matrices21.4Common Matrix Derivatives21.5Automatic Differentiation
  3. 22The Mathematics of Attention6 sections · 106m

    The linear algebra inside transformers

    22.1–22.6
    22.1Attention as Weighted Combination22.2Query, Key, Value: The QKV Framework22.3Scaled Dot-Product Attention22.4Multi-Head Attention22.5Positional Encoding22.6The Full Transformer Layer
  4. 23Numerical Linear Algebra5 sections · 89m

    Making linear algebra work on computers

    23.1–23.5
    23.1Floating Point and Numerical Error23.2Condition Numbers and Stability23.3Iterative Methods23.4Sparse Matrices23.5GPU Parallelism for Linear Algebra

Part VI

Applications

Computer vision, ML, and beyond

  1. 24Linear Algebra in Computer Vision6 sections · 111m

    Images as matrices and beyond

    24.1–24.6
    24.1Images as Matrices24.2Convolution as Matrix Multiplication24.3Image Transformations24.4Homography and Perspective24.5Optical Flow24.6Structure from Motion
  2. 25Linear Algebra in Machine Learning6 sections · 107m

    The mathematical core of ML algorithms

    25.1–25.6
    25.1Data as Matrices25.2Linear Models25.3Kernel Methods25.4Neural Network Layers25.5Embeddings and Representations25.6Recommender Systems
  3. 26Linear Algebra Across Fields6 sections · 103m

    Applications in graphs, signals, quantum, and more

    26.1–26.6
    26.1Graph Theory and Networks26.2Signal Processing and Fourier26.3Quantum Computing26.4Control Systems26.5Cryptography26.6Economics and Game Theory

134 sections. Begin with one.

Chapter 1 — The Geometric Universe — is where every reader starts.

In progress — 134 of 139 lessons published

5 more sections are still being written and are not part of this count.

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