Linear algebra¶
A book in 20 chapters about vectors, matrices and the spaces they live in: from arrows and systems of equations to vector spaces, eigenvalues, orthogonality, the spectral theorem, the singular value decomposition and Jordan form, and on to tensors, numerical methods and the open problems. Each chapter starts from a question, proves what it claims, works examples step by step, and ends with questions whose answers are folded underneath.
How to read it¶
- In order. Each chapter needs only the ones before it; the Next link at the end of every chapter walks you through.
- Levels inside a chapter. The box at the top of each chapter says which sections are beginner, intermediate, advanced or expert. On a first read, skip the harder sections and come back later.
- Little needed to start. Part I needs only school algebra. A few later chapters use calculus (derivatives and differential equations); the box says so, and the Calculus and analysis book is there.
- Pictures and computation. Linear algebra is both geometry and arithmetic. The chapters draw what a matrix does to the plane, and many show a few lines of Python that check the arithmetic.
- Where it is used. Statistics, probability, calculus in several variables, codes, graphics and machine learning all run on it; the chapters link to the other books where they meet.
Contents¶
| Part | Chapter | Level |
|---|---|---|
| I. Vectors and equations | 1. Vectors | beginner |
| 2. Systems of linear equations | beginner | |
| 3. Matrices | beginner to intermediate | |
| II. Vector spaces | 4. Vector spaces | intermediate |
| 5. Independence, bases and dimension | intermediate | |
| 6. Linear maps | intermediate | |
| 7. The four subspaces | intermediate | |
| III. Determinants and eigenvalues | 8. Determinants | intermediate |
| 9. Eigenvalues and eigenvectors | intermediate | |
| 10. Applications of eigenvalues | intermediate to advanced | |
| IV. Orthogonality | 11. Inner product spaces | intermediate to advanced |
| 12. Projections and least squares | intermediate to advanced | |
| 13. Orthogonal matrices and rotations | advanced | |
| V. Structure theorems | 14. The spectral theorem | advanced |
| 15. The singular value decomposition | advanced | |
| 16. Jordan form | advanced | |
| VI. Further and frontier | 17. Duality and multilinear algebra | advanced to expert |
| 18. Numerical linear algebra | advanced | |
| 19. Linear algebra in the world | advanced | |
| 20. Where linear algebra goes | expert |
The glossary defines every term, with a link to the chapter that teaches it.
Three ways in¶
- Starting out: chapters 1 to 3, then 8 and 9. You will be able to solve systems, multiply matrices and find eigenvalues, and know why the methods work.
- For a first university course: chapters 4 to 16 in order.
- For data and applications: chapters 9, 12, 14, 15 and 18 to 19, with the Statistics book beside you.