Preface
This book is for a reader who has met matrices and small linear systems informally and now needs the subject proved. It aims to take that reader from vector spaces over a field to the spectral theorem, the singular value decomposition and the first ideas of numerical linear algebra, with the theory and the computation kept side by side.
It can be read straight through, used as the text for a course, or opened as a reference. It is also meant to be a way of preparing for the linear algebra part of a preliminary or qualifying exam, which is why the exercises matter as much as the chapters. Each section is written in layers you can switch on and off (intuition, statement, proof, remarks), so a reader can stop at the depth they need. Each chapter has a plate you can drag and push on, and a script that checks the chapter’s examples and formulas in exact arithmetic. Every check has a control that is built to fail, so a check that cannot fail does not count.
Every chapter opens by drawing the object it studies and saying in one line what that object is, and the drawing moves: you change it and watch what stays fixed. The proofs come after the reader has seen the thing, not before.
The reader is assumed to bring seriousness and some practice with proofs. No earlier formal linear algebra is assumed.
What this volume is, and is notIt is a new book written in this series’ own words. Its scope follows the standard graduate syllabus, and for that scope it uses Golan, The Linear Algebra a Beginning Graduate Student Ought to Know (2nd ed., Springer, 2007), as a reference; of that book only the contents list and the preface have been read so far. Nothing here is copied from it. From Chapter 3, the page-addressed text is Lax, Linear Algebra and Its Applications (2nd ed., Wiley, 2007), Chapter 1 so far; Lay (5th ed.) is held as a computational companion and not yet cited. A Lean file exists for Chapter 3 but has not been compiled, so nothing here is claimed as machine-checked.
Contents
3Fields and vector spacesReal, complex and finite fields; the axioms; subspaces and quotients. Follows Lax Ch 1 (pp.2–8); Lean file compiled.written
4Basis and dimensionWhy dimension is well defined; the exchange lemma, complements, the quotient count, sums of subspaces. Follows Lax Ch 1 (pp.5–11); Lean file compiled.written
5Linear mapsKernel, image, rank–nullity; the corollaries; polynomial interpolation; the Jacobian as the linear map that best fits a nonlinear one (one worked example). Follows Lax Ch 3 (pp.19–22); Lean file compiled.written
6Matrices, systems and determinantsA matrix as a map in a basis, the product rule, elimination and rank, the determinant (det(BA) = det A det B, invertible iff det ≠ 0), Cramer’s rule, trace and similarity. Follows Lax Ch 4–5 (pp.32–56); Lean file compiled (nine statements, standard axioms only).written
7Eigenvalues and eigenvectorsThe eigenvalue equation, the characteristic polynomial, distinct eigenvalues and the Fibonacci numbers, the 2×2 classification, spectral mapping and Cayley–Hamilton. Follows Lax Ch 6 (pp.59–68); Lean file compiled (seven statements, standard axioms only). Jordan form and the minimal polynomial are not in this chapter.written
8Inner products and the spectral theoremSchwarz, Gram–Schmidt, projection and the closest point, least squares, quadratic forms, the spectral theorem for symmetric matrices. Follows Lax Ch 7 and 8 (pp.77–107); Lean file compiled (eight statements, standard axioms only). Hermitian and normal maps are not in this chapter.written
9Norms, singular values and conditioningThe size of a map, polar and singular value decompositions, the condition number and sensitivity of a linear system, the pseudoinverse and why the normal equations square the condition number, the Hilbert matrix. Follows Lax Ch 7, 8, 10 and 17 (pp.89–91, 119, 169–171, 246–248); Lean file compiled, ten of ten clean. Krylov methods, the QR algorithm and floating-point error models are not in this chapter.written
10Duality, bilinear forms, tensor productsThe dual space and dual basis, annihilators, the transpose as the dual map, a quadrature rule found in the dual space, bilinear forms (congruence versus similarity, signature), and the first tensor ideas (Kronecker product, pure tensors). Follows Lax Ch 2 (pp.13–18); Lean file written, not yet compiled. Sylvester's law is checked for 2×2 only; infinite-dimensional duals, exterior algebra and the universal property are not in this chapter.written
The order and the chapter titles can still change. Chapters 1 to 10 and both interludes exist. The book goes back and forth: where a chapter has a nonlinear counterpart (Strogatz is the working reference for that half), it is placed beside the linear result, with a label saying whether the link is a definition, an approximation or a derivation.
How this connects to the rest of the series
Book XXI depends on linear algebra at a narrow point: its theorem that two spiral sinks with different μ are not similar is a statement about matrices up to a change of basis, which Chapter 1 here introduces in the plane. That is a reading order, not a claim of shared structure. Where a later chapter here meets an earlier one of the series, the link will say whether it is the same definition, a formal analogy, or a derivation, and a connection without one of those labels will not be drawn.