A rough guide to linear algebra

Dongryul Kim

Web edition Updated August 25, 2026

About the book

A rough guide to linear algebra develops linear algebra through the abstract perspective of vector spaces and linear maps. The book is intended for readers who are comfortable with rigorous mathematical writing and want an abstract second look at linear algebra. This website and its downloadable PDF are maintained as a single rolling release.

  1. Preface
  2. 1Sets
    1. 1.1Sets and maps
    2. 1.2Products, coproducts, and sets of maps
    3. 1.3Fun with diagrams
    4. 1.4Equivalence classes
  3. 2Vector spaces
    1. 2.1Fields
    2. 2.2Vector spaces
    3. 2.3Matrices
    4. 2.4Products and direct sums
    5. 2.5Subspaces and quotients
    6. 2.6Vector spaces from linear maps
    7. 2.7Bases and dimension
    8. 2.8Dual spaces
    9. 2.9Linear algebra in combinatorics
  4. 3Multilinear algebra
    1. 3.1Bilinear maps and tensor products
    2. 3.2Symmetric and exterior algebras
    3. 3.3The determinant
    4. 3.4Computing the inverse matrix
  5. 4Linear algebra without division
    1. 4.1Commutative rings
    2. 4.2Modules
    3. 4.3Classification of finitely generated modules over a PID
    4. 4.4Frobenius and Jordan normal form
    5. 4.5Eigenvalues and eigenvectors
  6. 5Linear algebra over R and C
    1. 5.1A bit of analysis
    2. 5.2Inner products
    3. 5.3Operators on Hilbert spaces
    4. 5.4The spectral theorem
    5. 5.5Positivity of operators
  7. Epilogue