Linear Algebra: The Mathematics of Everything
The Mathematics of Everything
Start Reading →
40 chapters
~60 hours total
307 sections
1
Front Matter
4 chapters2
Part I — Vectors and Systems
7 chapters- Part I — Vectors and Systems
- What Is Linear Algebra? The Mathematics Hiding Inside Everything
- Vectors: Direction, Magnitude, and the Language of Space
- Systems of Linear Equations: The Problem That Started It All
- Gaussian Elimination and Row Reduction: The Algorithm That Solves Everything
- Vector Spaces: The Abstract Generalization (and Why It's Worth the Climb)
- Subspaces, Span, and Linear Independence: The Geometry of Solution Sets
3
Part II — Matrices as Transformations
7 chapters- Part II — Matrices as Transformations
- Matrices as Functions: What a Matrix DOES to Space
- Matrix Operations: Addition, Multiplication, and the Surprising Non-Commutativity
- The Inverse Matrix: Undoing a Transformation
- LU and PLU Decomposition: Efficient Solving at Scale
- The Determinant: Volume, Orientation, and Whether a Matrix Is Invertible
- Application: Computer Graphics — Rotation, Scaling, Projection, and Rendering in 3D
4
Part III — The Four Fundamental Subspaces
6 chapters- Part III — The Four Fundamental Subspaces
- Column Space and Null Space: What Ax = b Can and Cannot Reach
- Row Space, Left Null Space, and the Rank-Nullity Theorem
- Dimension, Basis, and Coordinates: How Many Numbers Do You Need?
- Change of Basis: Same Vector, Different Coordinate Systems
- Application: Linear Regression as Projection onto the Column Space
5
Part IV — Orthogonality
6 chapters- Part IV — Orthogonality
- Dot Products, Norms, and the Geometry of Angles in High Dimensions
- Orthogonal Projection: The Closest Point and the Least Squares Solution
- Gram-Schmidt and QR Decomposition: Making Orthonormal Bases
- Orthogonal Matrices and Rotations: Transformations That Preserve Distance
- Application: Fourier Series as Projection onto Orthogonal Basis Functions
6
Part V — Eigenvalues and Eigenvectors
8 chapters- Part V — Eigenvalues and Eigenvectors
- Eigenvalues and Eigenvectors: The Vectors That a Matrix Doesn't Rotate
- The Characteristic Polynomial and How to Find Eigenvalues
- Diagonalization: When a Matrix Reveals Its True Nature
- Complex Eigenvalues: Rotations in Disguise
- The Spectral Theorem: Symmetric Matrices Are Always Diagonalizable (and That's Profound)
- Positive Definite Matrices and Quadratic Forms: Energy, Curvature, and the Geometry of Optimization
- Application: PageRank — How Google Ranks the Internet with Eigenvectors
7
Part VI — Matrix Decompositions
5 chapters- Part VI — Matrix Decompositions
- The Singular Value Decomposition: The Most Important Factorization in Linear Algebra
- SVD Applications: Image Compression, Noise Reduction, and Dimensionality Reduction
- Principal Component Analysis: Finding the Directions That Matter in Data
- Application: Machine Learning — How Neural Networks, Embeddings, and Recommendation Systems Use Linear Algebra
8
Part VII — Advanced Topics
6 chapters- Part VII — Advanced Topics
- Inner Product Spaces: Generalizing Geometry Beyond Euclidean Space
- Linear Transformations and Abstract Vector Spaces: The Full Generalization
- Jordan Normal Form: When a Matrix Can't Be Diagonalized
- Linear Algebra Meets Calculus: The Matrix Exponential, Systems of Differential Equations, and Stability
- Numerical Linear Algebra: Stability, Conditioning, and How Computers Actually Do It
9
Part VIII — Synthesis
3 chapters10
Appendices
14 chapters- Topic Index
- Glossary
- Annotated Bibliography
- Answers to Selected Exercises
- Appendix A — Prerequisite Math Review
- Appendix B — Formula and Theorem Reference
- Appendix C — Python and numpy Setup Guide
- Appendix D — numpy / scipy.linalg Cheatsheet
- Appendix E — Proof Techniques
- Appendix F — Notation Reference
- Appendix G — Historical Notes
- Appendix H — Mapping to Strang's *Introduction to Linear Algebra*
- Appendix I — Mapping to Axler and to Boyd & Vandenberghe
- Appendix J — Applications Catalog
Explore Related Books
More open-access textbooks from our library
Advanced COBOL 40 chapters · ~67h Advanced Data Science 39 chapters · ~57h AI Ethics 39 chapters · ~82h AI Literacy 21 chapters · ~13h AI & ML for Business 40 chapters · ~80h AI Engineering 40 chapters · ~53h Algorithmic Addiction 40 chapters · ~71h American Government 40 chapters · ~77h Applied Psychology 40 chapters · ~52h Assembly Language 40 chapters · ~27h Blockchain & Crypto 40 chapters · ~68h Calculus 40 chapters · ~51h College Football Analytics 28 chapters · ~18h Creator Economy 41 chapters · ~57h Pattern Recognition 43 chapters · ~92h Data & Society 40 chapters · ~71h Data Viz with Python 35 chapters · ~53h Discrete Mathematics for Computer Science 40 chapters · ~75h Ethical Hacking 41 chapters · ~58h Fandom 44 chapters · ~71h History of Appalachia 42 chapters · ~69h How Humans Get Stuck 40 chapters · ~36h Handling Confrontation 40 chapters · ~80h How to Learn Anything 38 chapters · ~54h How Your House Works 40 chapters · ~66h IBM DB2 37 chapters · ~53h Intermediate COBOL 54 chapters · ~44h Intermediate Data Science 36 chapters · ~39h Intro CS Python 27 chapters · ~13h Intro to Data Science 36 chapters · ~54h Introductory Economics 40 chapters · ~12h Introductory Statistics 28 chapters · ~47h Learning COBOL 42 chapters · ~64h Prediction Markets 42 chapters · ~60h Metacognition 28 chapters · ~52h Media Literacy 41 chapters · ~81h NFL Analytics 28 chapters · ~16h Nuclear Physics 35 chapters · ~28h Organic Chemistry 40 chapters · ~21h Pascal Programming 40 chapters · ~43h Physics of Music 48 chapters · ~75h Political Analytics 41 chapters · ~67h Popular Psychology 40 chapters · ~21h Practical Philosophy 38 chapters · ~63h Basketball Analytics 31 chapters · ~30h Soccer Analytics 30 chapters · ~43h Propaganda 40 chapters · ~80h Python for Business 40 chapters · ~40h Quantum Mechanics 40 chapters · ~66h RegTech 40 chapters · ~59h The Science of Cooking 40 chapters · ~70h Science of Seduction 45 chapters · ~60h Sports Betting 42 chapters · ~63h Architecture of Surveillance 40 chapters · ~54h Science of Luck 40 chapters · ~72h Vibe Coding 42 chapters · ~58h Video Game Design 40 chapters · ~36h Why They Watch 40 chapters · ~48h Working with AI 42 chapters · ~58h