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

Chapter 1

Sets

Most of mathematics is grounded on the notion of sets. In this chapter, we quickly review the basic set theory we use throughout the book, assuming that the reader is familiar with most of the material.

Sections

  1. 1.1 Sets and maps
  2. 1.2 Products, coproducts, and sets of maps
  3. 1.3 Fun with diagrams
  4. 1.4 Equivalence classes
Previous Preface Next §1.1 Sets and maps
© Dongryul Kim Web edition: August 25, 2026 CC BY 4.0 Latest PDF GitHub repository